Exercise 744Number theory › Divisibility and gcdL1Added 2 Nov 2026

If $a,b$ are coprime, so are $a+b$ and $ab$

#coprime#sum#product

Statement

Let $a,b$ be coprime positive integers. Show that $\divisibility(a+b,ab)=1$ .

Hint

Let $p$ be a prime dividing both $a+b$ and $ab$ . Use Euclid's lemma.

Show solutionHide solution

Prime argument. Suppose a prime $p$ divides $a+b$ and $ab$ . Since $p\mid ab$ and $p$ is prime, $p\mid a$ or $p\mid b$ .

If $p\mid a$ , then from $p\mid a+b$ we get $p\mid b$ , contradicting $\divisibility(a,b)=1$ . Similarly if $p\mid b$ .

Conclusion. No prime divides both $a+b$ and $ab$ , so $\divisibility(a+b,ab)=1$ .