How many complex numbers a + ib, with a and b both integers, have modulus 5?

Step 1. The equation. $|a + ib| = 5$ means $a^2 + b^2 = 25$.

Step 2. The sizes. The only ways to write $25$ as a sum of two squares are $0 + 25$, $9 + 16$ and $16 + 9$.

Step 3. On the axes. $a = 0, b = \pm 5$ and $a = \pm 5, b = 0$ give $4$ numbers.

Step 4. Off the axes. $|a| = 3, |b| = 4$ gives $4$ sign choices, and $|a| = 4, |b| = 3$ gives $4$ more.