Mathematical_Recreations-Kraitchik-2e

76 Mathematical Recreations
Here are some problems on cyclic sets of numbers in the
decimal scale.
1. A number ends with the digit 2. H we move the 2
from the last place to the first, the new number is twice the
original. What is it?
Solution: Let N be the number and n the number of its
digits. Then
N-2
10 + 2. 1O n - 1 = 2N,
whence 19N = 2(lOn - 1). Hence n must be chosen
so that 19 divides IOn - 1. n = 18, 36, 54"", and N =
105263 157894736842, or any number formed by repeating
these digits. The fundamental sequence is the period in the
decimal expansion of ft.
2. A number begins with the digit 3. If this digit be
transposed to the last place, the new number is t of the original.
What is it?
Answer: 3103448275862068965517241379, or any
number formed by a finite number of repetitions of this sequence
of digits.
3. What number ending in 4 is multiplied by 4 when the
final digit is transposed to the first place?
Answer: 102564 is the smallest solution.
4. What number ending in 4 is doubled when the final
digit is transposed to the first position?
Answer: 210 526 315 789 473 684 is the smallest solution.
The number 142857 is the only number such that every
cyclic permutation of it is a multiple of it. This number, its
double, and numbers formed from either of them by an arbitrary
number of repetitions of its sequence of digits are the
only ones which generate more than one multiple of themselves
in the sequence of their cyclic permutations.