P-adic valuation of a product

Statement

Let $p$ be a prime number, and let $a$ and $b$ be nonzero integers. Then $\nu_p(a b) = \nu_p(a) + \nu_p(b)$, where $\nu_p$ is the p-adic valuation.

Remarks

With the convention $\nu_p(0) = \infty$, the formula holds for all integers $a$ and $b$. The proof uses Euclid's lemma. Write $a = p^i a'$ and $b = p^j b'$ with $p \nmid a'$ and $p \nmid b'$. Then $p \nmid a' b'$, so $\nu_p(a b) = i + j$. For example, $\nu_2(12 \cdot 10) = \nu_2(120) = 3 = 2 + 1$.

Explanations

L1 human

The factors $p$ in $a b$ come from $a$ or from $b$, and no new factor $p$ appears. So the number of factors $p$ in the product is the sum of the numbers of factors $p$ in $a$ and in $b$.