M is the smallest positive integer such that 2008*M is a perfect square and 2009*M is a perfect cube.

Analytically determine the remainder when M is divided by 25.

(In reply to Solution by Dej Mar)

I found something different (Dej Mar made a mistake considering the two numbers (square and cube) separately.)

Joining the two numbers, we have these factors:

(2^3 x 251) x ( 7^2 x 41)

So, to have a square and a cube, we need M = 2^3 x 7^4 x 251^3 x 41^2 =  510,588,495,274,648.

Edited to say that is the number Charlie found, too. I missed it.

Edited on October 6, 2008, 12:03 pm

Posted by pcbouhid on 2008-10-06 10:53:38