"Nathan claims that if you pick a two-digit number whose units digit is odd, but not 5, such as 37, and multiply it by some positive integer n and tell him the last two digits of your result that he can tell you the remainder when n is divided by 25. Is Nathan's claim possible? Why or why not?"
So, the units digit is odd and not $5$. That means $100$ and our number is coprime. I know that $100 = 2^25^2$, but from here I don't know what other conclusions I can make.