Solution

Show that the product of two integers of the form $6k+5$ is of the form $6k+1$.


Suppose $a=6k_1+5$ and $b=6k_2+5$ for some integers $k_1$ and $k_2$. Consider their product, $ab$: $$\begin{align}ab &= (6k_1+5)(6k_2+5)\\ &= 36k_1 k_2 + 30k_1 + 30k_2 + 25\\ &= 36k_1 k_2 + 30k_1 + 30k_2 + 24 + 1\\ &= 6(6k_1 k_2 + 5k_1 + 5k_2 + 4) + 1\end{align}$$ We know that $(6k_1 k_2 + 5k_1 + 5k_2 + 4)$ must be an integer due to the closure of integers under addition and multiplication. So, defining $k_3 = 6k_1 k_2 + 5k_1 + 5k_2 + 4$, we see that $ab$ can be written as $6k_3+1$. Thus, the product of two integers of the form $6k+5$ is of the form $6k+1$.