Exercise 1135Number theory › PrimesL1Added 3 Nov 2026

GCD of a product

#gcd#product#prime factorization

Statement

Show that $\gcd(ab,c)$ divides $\gcd(a,c)\gcd(b,c)$ .

Hint

Use prime factorizations: the exponent of $p$ in $\gcd(ab,c)$ is $\min(v_p(a)+v_p(b),v_p(c))$ .

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$p$ -adic. $v_p(\gcd(ab,c))=\min(v_p(a)+v_p(b),v_p(c))\le\min(v_p(a),v_p(c))+\min(v_p(b),v_p(c))=v_p(\gcd(a,c)\gcd(b,c))$ .

Conclusion. The inequality holds for each prime, so $\gcd(ab,c)\mid\gcd(a,c)\gcd(b,c)$ .

Example. $a=4$ , $b=6$ , $c=12$ : $\gcd(24,12)=12$ , $\gcd(4,12)\gcd(6,12)=4\cdot6=24$ . Divides indeed.