# More PreRMO or PRMO practice questions 2019

1. Find the remainder when $37^{100}$ is divided by 29.
2. Compute the last two digits of $9^{1500}$.
3. Prove for any integers a and b that $a^{2}+b^{2}$ never leaves a remainder of 3 when divided by 4.
4. Prove that the equation $a^{2}+b^{3}=3c^{2}$ has no solutions in non zero integers a, b and c.
5. Prove that the square of any odd integer always leaves a remainder of 1 when divided by 8.

More later,

Nalin Pithwa

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