Harmonic progression (mathematics)

The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are:

• 142857 × 3 = 428571 (shifts cyclically one place left)
• 142857 × 5 = 714285 (shifts cyclically one place right)
• 128205 × 4 = 512820 (shifts cyclically one place right)
• 076923 × 9 = 692307 (shifts cyclically two places left)

These specific integers, known as transposable integers, can be but are not always cyclic numbers. The characterization of such numbers can be done using repeating decimals (and thus the related fractions), or directly.

General

For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits. E.g. Template:Frac = 0.006993006993006993...

While the expression of a single series with vinculum on top is adequate, the intention of the above expression is to show that the six cyclic permutations of 006993 can be obtained from this repeating decimal if we select six consecutive digits from the repeating decimal starting from different digits.

This illustrates that cyclic permutations are somehow related to repeating decimals and the corresponding fractions.

The greatest common divisor (gcd) between any cyclic permutation of an m-digit integer and 10^m − 1 is constant. Expressed as a formula,

$\gcd (N,10^m-1)=\gcd (N_c,10^m-1),$

where N is an m-digit integer; and N_c is any cyclic permutation of N.

For example,

   gcd(091575, 999999) = gcd(32×52×11×37, 33×7×11×13×37)
                       = 3663
                       = gcd(915750, 999999)
                       = gcd(157509, 999999)
                       = gcd(575091, 999999)
                       = gcd(750915, 999999)
                       = gcd(509157, 999999)

If N is an m-digit integer, the number N_c, obtained by shifting N to the left cyclically, can be obtained from:

$N_c=10N-d(10^m-1),$

where d is the first digit of N and m is the number of digits.

This explains the above common gcd and the phenomenon is true in any base if 10 is replaced by b, the base.

The cyclic permutations are thus related to repeating decimals, the corresponding fractions, and divisors of 10^m−1. For examples the related fractions to the above cyclic permutations are thus:

• Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, and Template:Frac.

Reduced to their lowest terms using the common gcd, they are:

• Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, and Template:Frac.

That is, these fractions when expressed in lowest terms, have the same denominator. This is true for cyclic permutations of any integer.

Fraction method

Integral multiplier

An integral multiplier refers to the multiplier n being an integer:

1. An integer X shift right cyclically by k positions when it is multiplied by an integer n. X is then the repeating digits of Template:Frac, whereby F is F₀ = n 10^k − 1 (F₀ is coprime to 10), or a factor of F₀; excluding any values of F which are not more than n.
2. An integer X shift left cyclically by k positions when it is multiplied by an integer n. X is then the repeating digits of Template:Frac, whereby F is F₀ = 10^k - n, or a factor of F₀; excluding any values of F which are not more than n and which are not coprime to 10.

It is necessary for F to be coprime to 10 in order that Template:Frac is a repeating decimal without any preceding non-repeating digits (see multiple sections of Repeating decimal). If there are digits not in a period, then there is no corresponding solution.

For these two cases, multiples of X, i.e. (j X) are also solutions provided that the integer i satisfies the condition Template:Frac < 1. Most often it is convenient to choose the smallest F that fits the above. The solutions can be expressed by the formula:

$X=j\frac{10^p-1}{F}$

where p is a period length of Template:Frac; and F is a factor of F₀ coprime to 10.

E.g, F₀ = 1260 = 2² × 3² × 5 × 7. The factors excluding 2 and 5 recompose to F = 3² × 7 = 63. Alternatively, strike off all the ending zeros from 1260 to become 126, then divide it by 2 (or 5) iteratively until the quotient is no more divisible by 2 (or 5). The result is also F = 63.

To exclude integers that begin with zeros from the solutions, select an integer j such that Template:Frac > Template:Frac, i.e. j > Template:Frac.

There is no solution when n > F.

Fractional multiplier

An integer X shift left cyclically by k positions when it is multiplied by a fraction Template:Frac. X is then the repeating digits of Template:Frac, whereby F is F₀ = s 10^k - n, or a factor of F₀; and F must be coprime to 10.

For this third case, multiples of X, i.e. (j X) are again solutions but the condition to be satisfied for integer j is that Template:Frac < 1. Again it is convenient to choose the smallest F that fits the above.

The solutions can be expressed by the formula:

$X=js\frac{10^p-1}{F}$

where p is defined likewise; and F is made coprime to 10 by the same process as before.

To exclude integers that begin with zeros from the solutions, select an integer j such that Template:Frac > Template:Frac, i.e. j > Template:Frac.

Again if Template:Frac > 1, there is no solution.

Direct representation

The direct algebra approach to the above cases integral multiplier lead to the following formula:

1. $X=D\frac{10^m-1}{n10^k-1},$

   where m is the number of digits of X, and D, the k-digit number shifted from the low end of X to the high end of n X, satisfies D < 10^k.

   If the numbers are not to have leading zeros, then n 10^k − 1 ≤ D.

2. $X=D\frac{10^m-1}{10^k-n},$

   where m is the number of digits of X, and D, the k-digit number shifted from the high end of X to the low end of n X, satisfies:

   1. $D<\frac{10^k}{n}-1,$
   2. and the 10-part (the product of the terms corresponding to the primes 2 and 5 of the factorization) of 10^k − n divides D.

      The 10-part of an integer t is often abbreviated $\gcd (10^{\mathrm{\infty }},t).$

   If the numbers are not to have leading zeros, then 10^k − 1 ≤ D.

Cyclic permutation by multiplication

A long division of 1 by 7 gives:

        0.142857...
    7 ) 1.000000
         .7
          3
          28
           2
           14
            6
            56
             4
             35
              5
              49
               1

At the last step, 1 reappears as the remainder. The cyclic remainders are {1, 3, 2, 6, 4, 5}. We rewrite the quotients with the corresponding dividend/remainders above them at all the steps:

    Dividend/Remainders    1 3 2 6 4 5
    Quotients              1 4 2 8 5 7

and also note that:

• Template:Frac = 0.142857...
• Template:Frac = 0.428571...
• Template:Frac = 0.285714...
• Template:Frac = 0.857142...
• Template:Frac = 0.571428...
• Template:Frac = 0.714285...

By observing the remainders at each step, we can thus perform a desired cyclic permutation by multiplication. E.g.,

• The integer 142857, corresponding to remainder 1, permutes to 428571 when multiplied by 3, the corresponding remainder of the latter.
• The integer 142857, corresponding to remainder 1, permutes to 857142 when multiplied by 6, the corresponding remainder of the latter.
• The integer 857142, corresponding to remainder 6, permutes to 571428 when multiplied by Template:Frac; i.e. divided by 6 and multiplied by 5, the corresponding remainder of the latter.

In this manner, cyclical left or right shift of any number of positions can be performed.

Less importantly, this technique can be applied to any integer to shift cyclically right or left by any given number of places for the following reason:

• Every repeating decimal can be expressed as a rational number (fraction).
• Every integer, when added with a decimal point in front and concatenated with itself infinite times, can be converted to a fraction, e.g. we can transform 123456 in this manner to 0.123456123456..., which can thus be converted to fraction Template:Frac. This fraction can be further simplified but it will not be done here.
• To permute the integer 123456 to 234561, all one needs to do is to multiply 123456 by Template:Frac. This looks like cheating but if Template:Frac is a whole number (in this case it is not), the mission is completed.

Proof of formula for cyclical right shift operation

An integer X shift cyclically right by k positions when it is multiplied by an integer n. Prove its formula.

Proof

First recognize that X is the repeating digits of a repeating decimal, which always possesses cyclic behavior in multiplication. The integer X and its multiple n X then will have the following relationship:

1. The integer X is the repeating digits of the fraction Template:Frac, say d_pd_p-1...d₃d₂d₁, where d_p, d_p-1, ..., d₃, d₂ and d₁ each represents a digit and p is the number of digits.
2. The multiple n X is thus the repeating digits of the fraction Template:Frac, say d_kd_k-1...d₃d₂d₁d_pd_p-1...d_k+2d_k+1, representing the results after right cyclical shift of k positions.
3. F must be coprime to 10 so that when Template:Frac is expressed in decimal there is no preceding non-repeating digits otherwise the repeating decimal does not possesses cyclic behavior in multiplication.
4. If the first remainder is taken to be n then 1 shall be the (k + 1)^th remainder in the long division for Template:Frac in order for this cyclic permutation to take place.
5. In order that n × 10^k = 1 (mod F) then F shall be either F₀ = (n × 10^k - 1), or a factor of F₀; but excluding any values not more than n and any value having a nontrivial common factor with 10, as deduced above.

This completes the proof.

Proof of formula for cyclical left shift operation

An integer X shift cyclically left by k positions when it is multiplied by an integer n. Prove its formula.

Proof

First recognize that X is the repeating digits of a repeating decimal, which always possesses a cyclic behavior in multiplication. The integer X and its multiple n X then will have the following relationship:

1. The integer X is the repeating digits of the fraction Template:Frac, say d_pd_p-1...d₃d₂d₁ .
2. The multiple n X is thus the repeating digits of the fraction Template:Frac, say d_p-kd_p-k-1...d₃d₂d₁d_pd_p-1...d_p-k+1,

which represents the results after left cyclical shift of k positions.

1. F must be coprime to 10 so that Template:Frac has no preceding non-repeating digits otherwise the repeating decimal does not possesses cyclic behavior in multiplication.
2. If the first remainder is taken to be 1 then n shall be the (k + 1)^th remainder in the long division for Template:Frac in order for this cyclic permutation to take place.
3. In order that 1 × 10^k = n (mode F) then F shall be either F₀ = (10^k -n), or a factor of F₀; but excluding any value not more than n, and any value having a nontrivial common factor with 10, as deduced above.

This completes the proof. The proof for non-integral multiplier such as Template:Frac can be derived in a similar way and is not documented here.

Shifting an integer cyclically

The permutations can be:

• Shifting right cyclically by single position (parasitic numbers);
• Shifting right cyclically by double positions;
• Shifting right cyclically by any number of positions;
• Shifting left cyclically by single position;
• Shifting left cyclically by double positions; and
• Shifting left cyclically by any number of positions

Parasitic numbers

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When a parasitic number is multiplied by n, not only it exhibits the cyclic behavior but the permutation is such that the last digit of the parasitic number now becomes the first digit of the multiple. For example, 102564 x 4 = 410256. Note that 102564 is the repeating digits of Template:Frac and 410256 the repeating digits of Template:Frac.

Shifting right cyclically by double positions

An integer X shift right cyclically by double positions when it is multiplied by an integer n. X is then the repeating digits of Template:Frac, whereby Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

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Summary of results

The following multiplication moves the last two digits of each original integer to the first two digits and shift every other digits to the right:

Multiplier n	Solution	Represented by	Other Solutions
2	0050251256 2814070351 7587939698 4924623115 5778894472 3618090452 2613065326 6331658291 4572864321 608040201	Template:Frac x 2 = Template:Frac period = 99 i.e. 99 repeating digits.	Template:Frac, Template:Frac, ..., Template:Frac
3	0033444816 0535117056 8561872909 6989966555 1839464882 9431438127 090301	Template:Frac x 3 = Template:Frac period = 66 299 = 13×23	Template:Frac, Template:Frac, ..., Template:Frac some special cases are illustrated below
3	076923	Template:Frac x 3 = Template:Frac period = 6	Template:Frac, Template:Frac, Template:Frac
3	0434782608 6956521739 13	Template:Frac x 3 = Template:Frac period = 22	Template:Frac, Template:Frac, ..., Template:Frac
4	0025062656 64160401	Template:Frac x 4 = Template:Frac period = 18 399 = 3×7×19	Template:Frac, Template:Frac, ..., Template:Frac some special cases are illustrated below
4	142857	Template:Frac x 4 = Template:Frac period = 6	-
4	0526315789 47368421	Template:Frac x 4 = Template:Frac period = 18	Template:Frac, Template:Frac, Template:Frac
5	(a cyclic number with a period of 498)	Template:Frac x 5 = Template:Frac 499 is a full reptend prime	Template:Frac, Template:Frac, ..., Template:Frac

Note that:

• 299 = 13 x 23, and the period of Template:Frac is accurately determined by the formula, LCM(6, 22) = 66, according to Repeating decimal#Generalization.
• 399 = 3 x 7 x 19, and the period of Template:Frac is accurately determined by the formula, LCM(1, 6, 18) = 18.

There are many other possibilities.

Shifting left cyclically by single position

Problem: An integer X shift left cyclically by single position when it is multiplied by 3. Find X.

Solution: First recognize that X is the repeating digits of a repeating decimal, which always possesses some interesting cyclic behavior in multiplications. The integer X and its multiple then will have the following relationship:

• The integer X is the repeating digits of the fraction Template:Frac, say ab***.
• The multiple is thus the repeating digits of the fraction Template:Frac, say b***a.
• In order for this cyclic permutation to take place, then 3 shall be the next remainder in the long division for Template:Frac. Thus Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
  
  Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
  
  There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
  
  This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
  
  A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
  
  If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
  
  Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. shall be 7 as 1 × 10 ÷ 7 gives remainder 3.

This yields the results that:

X = the repeating digits of Template:Frac

=142857, and

the multiple = 142857 × 3 = 428571, the repeating digits of Template:Frac

The other solution is represented by Template:Frac x 3 = Template:Frac:

• 285714 x 3 = 857142

There are no other solutions because:

• Integer n must be the subsequent remainder in a long division of a fraction Template:Frac. Given that n = 10 - F, and F is coprime to 10 in order for Template:Frac to be a repeating decimal, then n shall be less than 10.
• For n = 2, F must be 10 - 2 = 8. However Template:Frac does not generate a repeating decimal, similarly for n = 5.
• For n = 7, F must be 10 - 7 = 3. However 7 > 3 and Template:Frac = 2.333 > 1 and does not fit the purpose.
• Similarly there is no solution for any other integer of n less than 10 except n = 3.

However, if the multiplier is not restricted to be an integer (though ugly), there are many other solutions from this method. E.g., if an integer X shift right cyclically by single position when it is multiplied by Template:Frac, then 3 shall be the next remainder after 2 in a long division of a fraction Template:Frac. This deduces that F = 2 x 10 - 3 = 17, giving X as the repeating digits of Template:Frac, i.e. 1176470588235294, and its multiple is 1764705882352941.

The following summarizes some of the results found in this manner:

Multiplier Template:Frac	Solution	Represented by	Other Solutions
Template:Frac	105263157894736842	Template:Frac × Template:Frac = Template:Frac A 2-parasitic number	Other 2-parasitic numbers: Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
Template:Frac	1176470588235294	Template:Frac × Template:Frac = Template:Frac	Template:Frac, Template:Frac, Template:Frac, Template:Frac
Template:Frac	153846	Template:Frac × Template:Frac = Template:Frac	-
Template:Frac	18	Template:Frac × Template:Frac = Template:Frac	-
Template:Frac	1304347826086956521739	Template:Frac × Template:Frac = Template:Frac	Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
Template:Frac	190476	Template:Frac × Template:Frac = Template:Frac	-

Shifting left cyclically by double positions

An integer X shift left cyclically by double positions when it is multiplied by an integer n. X is then the repeating digits of Template:Frac, whereby Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. = 10² - n, or a factor of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.; excluding values of Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. for which Template:Frac has a period length dividing 2 (or, equivalently, less than 3); and F must be coprime to 10.

Most often it is convenient to choose the smallest Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. that fits the above.

Summary of results

The following summarizes some of the results obtained in this manner, where the white spaces between the digits divide the digits into 10-digit groups:

Multiplier n	Solution	Represented by	Other Solutions
2	142857	Template:Frac × 2 = Template:Frac	Template:Frac, Template:Frac
3	0103092783 5051546391 7525773195 8762886597 9381443298 9690721649 4845360824 7422680412 3711340206 185567	Template:Frac x 3 = Template:Frac	Template:Frac, Template:Frac, Template:Frac, Template:Frac, ...., Template:Frac, Template:Frac
4	No solution	-	-
5	0526315789 47368421	Template:Frac x 5 = Template:Frac	Template:Frac, Template:Frac
6	0212765957 4468085106 3829787234 0425531914 893617	Template:Frac x 6 = Template:Frac	Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
7	0322580645 16129	Template:Frac x 7 = Template:Frac	Template:Frac, Template:Frac, Template:Frac Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
8	0434782608 6956521739 13	Template:Frac x 8 = Template:Frac	Template:Frac
9	076923	Template:Frac x 9 = Template:Frac	Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
10	No solution	-	-
11	0112359550 5617977528 0898876404 4943820224 7191	Template:Frac x 11 = Template:Frac	Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
12	No solution	-	-
13	0344827586 2068965517 24137931	Template:Frac x 13 = Template:Frac	Template:Frac Template:Frac, Template:Frac, Template:Frac, Template:Frac, Template:Frac
14	0232558139 5348837209 3	Template:Frac x 14 = Template:Frac	Template:Frac, Template:Frac
15	0588235294117647	Template:Frac x 15 = Template:Frac	-

References

• P. Yiu, k-right-transposable integers, k-left-transposable integers Chap.18.1, 18.2 pp. 168/360 in 'Recreational Mathematics', http://www.math.fau.edu/yiu/RecreationalMathematics2003.pdf
• C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford University Press UK, 2000.

• Template:SloanesRef

• Gardner, Martin. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments From Scientific American. New York: The Mathematical Association of America, 1979. pp. 111–122.

• Kalman, Dan; 'Fractions with Cycling Digit Patterns' The College Mathematics Journal, Vol. 27, No. 2. (Mar., 1996), pp. 109–115.

• Leslie, John. "The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of ....", Longman, Hurst, Rees, Orme, and Brown, 1820, ISBN 1-4020-1546-1

• Wells, David; "The Penguin Dictionary of Curious and Interesting Numbers", Penguin Press. ISBN 0-14-008029-5