Transposable integer
   HOME

TheInfoList



OR:

The digits of some specific integers
permute In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
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 number A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 :142 ...
s. The characterization of such numbers can be done using
repeating decimals A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
(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. = 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 permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
s 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\left(N,10^m-1\right)=\gcd\left(N_c,10^m-1\right), where ''N'' is an ''m''-digit integer; and ''Nc'' 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 ''Nc'', obtained by shifting ''N'' to the left cyclically, can be obtained from: :N_c = 10 N - d\left(10^m-1\right), \, 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: *, , , , , and . Reduced to their lowest terms using the common gcd, they are: *, , , , , and . That is, these fractions when expressed
in lowest terms An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction (mathematics), fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative nu ...
, 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: #An integer ''X'' shift right cyclically by ''k'' positions when it is multiplied by an integer ''n''. ''X'' is then the repeating digits of , whereby ''F'' is ''F''0 = ''n'' 10''k'' − 1 (''F''0 is
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to 10), or a factor of ''F''0; excluding any values of ''F'' which are not more than ''n''. #An integer ''X'' shift left cyclically by ''k'' positions when it is multiplied by an integer ''n''. ''X'' is then the repeating digits of , whereby ''F'' is ''F''0 = 10''k'' - ''n'', or a factor of ''F''0; excluding any values of ''F'' which are not more than ''n'' and which are not
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to 10. It is necessary for F to be coprime to 10 in order that is a repeating decimal without any preceding non-repeating digits (see multiple sections of
Repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
). 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 < 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 :where ''p'' is a period length of ; and ''F'' is a factor of ''F''0 coprime to 10. :E.g, ''F''0 = 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 > , i.e. ''j'' > . 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 . ''X'' is then the repeating digits of , whereby ''F'' is ''F''0 = s 10''k'' - ''n'', or a factor of ''F''0; 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 < 1. Again it is convenient to choose the smallest ''F'' that fits the above. The solutions can be expressed by the formula: :X = j s \frac :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 > , i.e. ''j'' > . Again if > 1, there is no solution.


Direct representation

The direct algebra approach to the above cases integral multiplier lead to the following formula: #X=D \frac , #: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''. #X = D \frac , #: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: #:#D<\frac n - 1, #:#and the 10-part (the product of the terms corresponding to the primes 2 and 5 of the
factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
) of 10''k'' − ''n'' divides ''D''. #:#:The 10-part of an integer ''t'' is often abbreviated \operatorname\left(10^\infty,t\right). #: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 . 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: * = 0.142857... * = 0.428571... * = 0.285714... * = 0.857142... * = 0.571428... * = 0.714285... By observing the remainders at each step, we can thus perform a desired
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
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 ; 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 . 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 . This looks like cheating but if 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 A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
, which always possesses cyclic behavior in multiplication. The integer ''X'' and its multiple ''n X'' then will have the following relationship: #The integer ''X'' is the repeating digits of the fraction , say ''dpdp-1...d3d2d1'', where ''dp'', ''dp-1'', ..., ''d3'', ''d2'' and ''d1'' each represents a digit and ''p'' is the number of digits. #The multiple ''n X'' is thus the repeating digits of the fraction , say ''dkdk-1...d3d2d1dpdp-1...dk+2dk+1'', representing the results after right cyclical shift of ''k'' positions. #''F'' must be coprime to 10 so that when is expressed in decimal there is no preceding non-repeating digits otherwise the repeating decimal does not possess cyclic behavior in multiplication. #If the first remainder is taken to be ''n'' then ''1'' shall be the (''k'' + 1)st remainder in the long division for in order for this cyclic permutation to take place. #In order that ''n'' × 10''k'' = 1 (mod ''F'') then ''F'' shall be either ''F''0 = (''n'' × 10k - 1), or a factor of ''F''0; 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 A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
, which always possesses a cyclic behavior in multiplication. The integer ''X'' and its multiple ''n X'' then will have the following relationship: #The integer ''X'' is the repeating digits of the fraction , say ''dpdp-1...d3d2d1'' . #The multiple ''n X'' is thus the repeating digits of the fraction , say ''dp-kdp-k-1...d3d2d1dpdp-1...dp-k+1'', which represents the results after left cyclical shift of ''k'' positions. #''F'' must be coprime to 10 so that has no preceding non-repeating digits otherwise the repeating decimal does not possesses cyclic behavior in multiplication. #If the first remainder is taken to be 1 then ''n'' shall be the (''k'' + 1)st remainder in the long division for in order for this cyclic permutation to take place. #In order that 1 × 10k = ''n'' (mode ''F'') then ''F'' shall be either ''F''0 = (10k -''n''), or a factor of ''F''0; 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 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 An ''n''-parasitic number (in base 10) is a positive natural number which, when multiplied by ''n'', results in movement of the last digit of its decimal representation to its front. Here ''n'' is itself a single-digit positive natural number. In ...
); *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

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 and 410256 the repeating digits of .


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 , whereby = ''n'' × 102 - 1; or a factor of it; but excluding values for which has a period length dividing 2 (or, equivalently, less than 3); and must be coprime to 10. Most often it is convenient to choose the smallest that fits the above.


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: Note that: *299 = 13 x 23, and the period of 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 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 A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if an ...
, 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 , say ''ab***''. *The multiple is thus the repeating digits of the fraction , say ''b***a''. *In order for this cyclic permutation to take place, then 3 shall be the next remainder in the long division for . Thus shall be 7 as 1 × 10 ÷ 7 gives remainder 3. This yields the results that: :''X'' = the repeating digits of ::=142857, and :the multiple = 142857 × 3 = 428571, the repeating digits of The other solution is represented by x 3 = : *285714 x 3 = 857142 There are no other solutions P. Yiu, k-right-transposable integers, Chap.18.1 'Recreational Mathematics' because: *Integer ''n'' must be the subsequent remainder in a long division of a fraction . Given that n = 10 - F, and F is coprime to 10 in order for to be a repeating decimal, then ''n'' shall be less than 10. *For ''n'' = 2, F must be 10 - 2 = 8. However does not generate a repeating decimal, similarly for ''n'' = 5. *For ''n'' = 7, F must be 10 - 7 = 3. However 7 > 3 and = 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 , then 3 shall be the next remainder after 2 in a long division of a fraction . This deduces that F = 2 x 10 - 3 = 17, giving ''X'' as the repeating digits of , i.e. 1176470588235294, and its multiple is 1764705882352941. The following summarizes some of the results found in this manner:


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 , whereby is = 102 - n, or a factor of ; excluding values of for which 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 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:


Other bases

In
duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wri ...
system, the transposable integers are: (using inverted two and three for ten and eleven, respectively) Note that the “Shifting left cyclically by single position” problem has no solution for the multiplier less than 12 except 2 and 5, the same problem in decimal system has no solution for the multiplier less than 10 except 3.


Notes


References

*P. Yiu, k-right-transposable integers, k-left-transposable integers Chap.18.1, 18.2 pp. 168/360 in 'Recreational Mathematics', https://web.archive.org/web/20090901180500/http://math.fau.edu/Yiu/RecreationalMathematics2003.pdf * C. A. Pickover, ''Wonders of Numbers'', Chapter 28,
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
UK, 2000. * *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, * Wells, David; ''"
The Penguin Dictionary of Curious and Interesting Numbers ''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, a ...
"'', Penguin Press. {{Classes of natural numbers Base-dependent integer sequences