Question

Prove that any integer $a\ge 7$ is the sum of two relatively prime integers. (We say two integers $m$ and $n$ are relatively prime if their HCF (i.e., GCD) is $1$)

Answer 1

Let $a$ be any integer.

When $a$ is odd then $a=p+q$ where either $p$ is even or $q$ is even.

If $p$ is even then $q$ is odd and hence $p$ and $q$ are relatively prime to each other. Similarly, vice versa.

Therefore, $GCD(p,q)=1$

Hence, any integer $n\ge 7$ is the sum of two relatively prime integers.