Exercise 932Number theory › Divisibility and gcdL1Added 3 Nov 2026

p-adic valuation

#p-adic valuation#divisibility#properties

Statement

For a prime $p$ and $n\in\mathbb{Z}\setminus\{0\}$ , define $v_p(n)$ as the exponent of $p$ in the prime factorization of $n$ . Show that:
1. $v_p(mn)=v_p(m)+v_p(n)$ ;
2. $v_p(m+n)\ge\min(v_p(m),v_p(n))$ , with equality if $v_p(m)\neq v_p(n)$ .

Hint

For (2), factor out $p^{\min}$ from both terms.

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Solution

1. Prime factorization of $mn$ combines the factorizations of $m$ and $n$ , and the exponents add.

2. Let $k=\min(v_p(m),v_p(n))$ . Write $m=p^km'$ , $n=p^kn'$ with $p\nmid m'$ or $p\nmid n'$ (at least one). Then $m+n=p^k(m'+n')$ , so $v_p(m+n)\ge k$ . If $v_p(m)\neq v_p(n)$ , exactly one of $m',n'$ is divisible by $p$ , so $m'+n'\not\equiv0\pmod p$ , giving $v_p(m+n)=k$ .

Conclusion. Both properties hold; $v_p$ behaves like a "non-Archimedean absolute value".

p-adic valuation

Statement

Solution

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