Find the number of positive integers that divide (10)^999 but not (10)^998.

10^999 = 5^999 * 2^999
10^998 = 5^998 * 2^998

So, every solution will either have one more 5 or one more 2 than 10^998.

if there is one extra 5 there can be 1000 (0 to 999) 2s and if there is one exra 2 there can be 999 (0 and 2 to 999 because we already counted it if there was one 5), 1000 + 999 = 1999.

The first time i looked at this i forgot to count for the fact that there could be one extra 5 and no 2s at all, but once i counted that i got the correct answer.

(Tip: looking at smaller numbers helps! I looked at 10^2 and 10^3 to check myself.)

Posted by Andy on 2006-05-18 18:54:41