The factorization of 2008 is 2^{3} x 251, thus a perfect square of 2008*K, where K is a positive integer, requires another factor of 2 and 251, i.e. K = 2^{4} x 251^{2}.
The factorization of 2009 is 7^{2} x 41, thus a perfect cube of 2009*L, where L is a positive integer, requires another factor of 7 and two more of 41, i.e., L = 7^{3} x 41^{3}.
Now if M must have the factors 2 x 7 x 251 x 41^{2} and 2008*M is a perfect square and 2009*M is a perfect cube, then two more factors of 2 and 251 and three more factors of 7 are needed with the composite factors of both K and L, thus, the smallest value of M would need to have the factors 2^{3} x 7^{4} x 41^{2} x 251^{3}. These factors equate to 510588495274648.
To find the remainder of M divided by 25, as any number being a multiple of 100 is also a multiple of 25, we only need to take the last two digits, 48, modulo 25. (48  25 = 23) The result being 23, thus the remainder of M/25 is 23/25.
Edited on October 6, 2008, 11:46 pm

Posted by Dej Mar
on 20081006 01:12:35 