Exercise 909Number theory › Divisibility and gcdL1Added 3 Nov 2026

Coprime product

#coprime#product#divisibility

Statement

Let $a,b$ be coprime and both divide $n$ . Show that $ab\mid n$ .

Hint

Use the prime factorization of $n$ and the fact that $a$ and $b$ have disjoint prime sets.

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Solution

Prime factorization. Write $n=\prod p^{\alpha_p}$ . Since $a\mid n$ and $b\mid n$ , the exponent of $p$ in $a$ and $b$ is at most $\alpha_p$ . Since $\gcd(a,b)=1$ , at most one of $a,b$ contains $p$ .

Conclusion. The exponent of $p$ in $ab$ is at most $\alpha_p$ , so $ab\mid n$ .

Application. If $a\mid n$ , $b\mid n$ , $\gcd(a,b)=1$ , then $ab\mid n$ .

Coprime product

Statement

Solution

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