EXERCISE

If $a$ and $b$ are two odd positive integers such that $a>b,$ then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even.

Prove that the product of two consecutive positive integers is divisible by $2$.

Prove that the product of three consecutive positive integers is divisible by $6.$

Prove that the square of any positive integer is of the form $4q$ or $4q+1$ for some integer $q$.

Prove that the square of an odd positive integer is of the form $8q+1,$ for some integer $q$.

Prove that the square of any positive integer is of the form $5q, 5q+1$ or $5q+4$ for some integer $q.$

Prove that if a positive integer is of the form $6q+5$, then it is of the form $3q+2$ for some integer $q,$ but not conversely.

Show that the square of an odd integer is of the form $8q+1,$ for some integer $q.$