2008 = 2^3 * 251
2009 = 7^2 * 41
m needs to have, as factors, an odd power of 2 and an odd power of 251, and they need be multiples of 3, so the power of each needs be 3 to be a minimum.
It also needs to have an even power of 7 such that two more than this power is a multiple of 3; 4 will do. And an even power of 41 such that the power plus 1 will be a multiple of 3; 2 will do.
So m = 2^3 * 251^3 * 7^4 * 41^2.
Mod 25:
2^3 = 8
251^3 = 1
7^4 = (1)^2 = 1
41^2 = 16^2 = 6
Product is 48 = 23
Verify:
m comes out to be 510588495274648, which is 23 mod 25.

Posted by Charlie
on 20081005 14:19:47 