## Solution to Problem #3

Since the first digit of the number is 1, the number can be writtten as
10^m + x, where x is an m digit number. Transferring the 1 to the end of
the number gives us the number 10 x + 1. We must therefore solve the
equation 10 x + 1 = 3(10^m + x), or equivalently 7 x + 1 = 3*10^m. In
other words, the remainder upon dividing 3*10^m by 7 must be 1. We can
either find the smallest m by trial and error or perform the following
long division until a remainder of 1 is obtained:

42857
_____________________
7) 3000000.....
28
--
20
14
--
60
56
--
40
35
--
50
49
--
1

So x = 42857 does the trick and the number we seek is 142857.

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