AIME I 2026 (Problem 8)
Let be the number of positive integer divisors of that leave a remainder of upon division by . Find the remainder when is divided by .
Prime factorize .
The positive divisors of are where
Therefore,
should be even.
should be odd.
There are possible values for . Since is even, has possible values. Since is odd, has possible values. Finally, has possible values.
Since the problem wants the remainder when is divided by ,This is not ordinary long multiplication.
It is a shortcut that keeps only the last three digits during the multiplication because only the remainder modulo is needed.
It is a shortcut that keeps only the last three digits during the multiplication because only the remainder modulo is needed.
244