UVa 10831

10831 - Gerg's Cake

• http://acm.uva.es/p/v108/10831.html

Summary

Given a and p, can a square cake be divided into a + n * p equal sized pieces?

Explanation

You have to test whether there is a solution to x² = a + n * p, where n is an integer. Taking everything modulo p, we get $x^2 \equiv a \pmod{p}$. Now we use a trick to get to Fermat's Little Theorem: we take everything to the power (p - 1) / 2, so we get $x^{p-1}=a^{(p-1)/2} \equiv 1 \pmod{p}$. So we only have to check whether $a^{(p-1)/2}\equiv 1 \pmod{p}$. If it is, there is a solution and otherwise there isn't. This can easily be calculated in O(logp).

Gotcha's

There are a few special cases, for example $a \equiv 0 \pmod{p}$, p = 1 and p = 2.

Input

1 3
1024 17
2 101
0 1
-1 -1

Output

Yes
Yes
No
Yes