For positive integers $a$ and $b$ and every prime $p$: $v_p(\gcd(a,b)) = \min\{v_p(a), v_p(b)\}$ and $v_p(\operatorname{lcm}(a,b)) = \max\{v_p(a), v_p(b)\}$; so the exponent-wise greatest common divisor is the $\gcd$ of the divisibility page and not a second notion

Let $a, b \in \mathbb{Z}$ with $a \ge 1$ and $b \ge 1$, write $g := \gcd(a,b)$ (Common divisor, and the greatest common divisor $\gcd(a,b)$, with the convention $\gcd(0,0) := 0$) and $\ell := \operatorname{lcm}(a,b)$ (Common multiple, and the least common multiple $\operatorname{lcm}(a,b)$, taken to be $0$ when $a = 0$ or $b = 0$), and let $p$ be any prime (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$). Then $g \ge 1$ and $\ell \ge 1$, so all four valuations below are defined (The $p$-adic valuation $v_p(a)$ of a nonzero integer: the greatest $k \in \mathbb{N}$ with $p^{k} \mid a$), and

1. $v_p(g) = \min\{\, v_p(a),\, v_p(b) \,\}$;
2. $v_p(\ell) = \max\{\, v_p(a),\, v_p(b) \,\}$,

the minimum and maximum being taken in $\mathbb{N}$, whose order is total ($\le$ is a linear order on $\mathbb{N}$).

3. The exponent recipe does not define a second notion. If $d \ge 1$ satisfies $v_p(d) = \min\{v_p(a),v_p(b)\}$ for every prime $p$, then $d = \gcd(a,b)$; and if $d \ge 1$ satisfies $v_p(d) = \max\{v_p(a),v_p(b)\}$ for every prime $p$, then $d = \operatorname{lcm}(a,b)$.

• This is the agreement item for the two pages. The exponent-wise recipe $\min$ for $\gcd$ and $\max$ for $\operatorname{lcm}$ is a computation of the $\gcd$ and $\operatorname{lcm}$ defined on the divisibility page, not a rival definition: clause 3 says any positive integer with the right valuations is that $\gcd$, respectively that $\operatorname{lcm}$. Nothing here redefines either symbol.

• The identity $\min + \max =$ sum recovers the product formula. Adding clauses 1 and 2 gives $v_p(g) + v_p(\ell) = v_p(a) + v_p(b) = v_p(ab)$ for every $p$, which is $g\ell = |ab|$ of Every common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a,b)$, and $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = |ab|$ read through valuations. That is a consistency check, and in this proof the implication was used in the other direction: the product formula is what pins $v_p(\ell)$ from above. The independent bound $M_p \le v_p(\ell)$, from $a \mid \ell$ and $b \mid \ell$, also follows from [L2] and [L4] and agrees.

• Both arguments are restricted to positive integers, and the restriction is not cosmetic. $v_p(0)$ is undefined (The $p$-adic valuation $v_p(a)$ of a nonzero integer: the greatest $k \in \mathbb{N}$ with $p^{k} \mid a$), so the pair $(0,0)$ — the one where the $\gcd$ convention lives — is out of scope entirely, and a negative argument is handled first by $\gcd(a,b) = \gcd(|a|,|b|)$ ($\gcd$ is symmetric and unchanged by signs: $\gcd(a,b) = \gcd(b,a) = \gcd(|a|,|b|)$; moreover $\gcd(a,0) = |a|$, $\gcd(a,1) = 1$, $\gcd(a,a) = |a|$, and $\gcd(a,b) \ge 1$ unless $a = b = 0$).

• The minimum and maximum are taken in $\mathbb{N}$, through the totality of its order ($\le$ is a linear order on $\mathbb{N}$), and not through the library's Maximum and minimum of a set, which is stated for real numbers. Valuations are natural numbers and there is no reason to leave $\mathbb{N}$.