Divisibility

For integers $a$ and $b$, one says that $a$ divides $b$, written $a\mid b$, if there exists an integer $c$ such that $b=ac$. Then $a$ is a divisor of $b$, and $b$ is a multiple of $a$.

When $a,b>0$, divisibility can be tested prime by prime: $a\mid b$ exactly when $v_p(a)\le v_p(b)$ for every prime $p$, where $v_p$ denotes the $p$-adic valuation.