Solution

Prove that the product of two numbers that take the form $5k+3$, where $k$ is some integer, takes the form $5k+4$.


Suppose $a$ and $b$ are numbers of the form $5k+3$, where $k$ is some integer. Then we can find integers $k_1$ and $k_2$, such that

$$\begin{array}{rcl} ab &=& (5k_1 + 3)(5k_2 + 3)\\ &=& 25k_1 k_2 + 15k_1 + 15k_2 + 9\\ &=& 25k_1 k_2 + 15k_1 + 15k_2 + 5 + 4\\ &=& 5(5k_1 k_2 + 3k_1 + 3k_2 + 1) + 4 \end{array}$$

We know $5k_1 k_2 + 3k_1 + 3k_2 + 1$ is an integer, by the closure of the integers under addition and multiplication. Hence, $ab$ takes the form $5k+4$ where $k$ is an integer.

QED.