Exercise 9. Determine the number of integers $1 \leqslant a \leqslant 1000$ such that $\left(a^{2}+1\right)\left(a^{2}+5\right)$ is divisible by 4.
Solution to exercise 9 If $a$ is even then $\left(a^{2}+1\right)\left(a^{2}+5\right)$ is odd and therefore not divisible by 4. If $a$ is odd then $a^{2}+1$ and $a^{2}+5$ are even so their product is divisible by 4. Therefore, there are 500 integers that satisfy the condition.