Find the smallest value of hexadecimal positive integer N which becomes 4*N when the last digit of N is moved to be the first. For example, turning (2345)_{16} into (5234)_{16}. How about 2*N? 7*N? 8*N?

Let the sought after N = Ak, where Ak represents a string of Hex digits.

k is a single Hex digit and A is a string of length m.

Then, 7*(A*16 + k) = k16^m + A

Solving, gives us A = k(16^m - 7)/111

Well, 111 = 3*37, so (16^m - 7) mod 37 must equal 0.

Put a little differently, 16^m = 7, mod 37

A tiny bit of Excel work shows that m = 8 (or 17 or 26).

(16^8 - 7)/37 = 116,080,197 base 10, so that is a candidate for our minimum A.

Happily, it is also divisible by 3, so k need not necessarily be a multiple of 3.

116080197 base 10 = 6EB3E45 base 16, which is only 7 digits.

2*116080197 is also going to be only 7 digits

So our minimum A = 3*116080197 = 348,240,591 base 10 = 14C1BACF base 16

Our initial A was equal to 3*(16^m - 7)/111, but it was not the required 8 digits.

This A = 9*(16^m - 7)/111, so k = 9.

Checking verifies this:

**N = 14C1BACF9** base 16 = 5,571,849,465

7 * 5,571,849,465 = 39,002,946,255 = 914C1BACF base 16

*Edited on ***December 23, 2014, 8:59 pm**