Product of GCD and LCM/Proof 4

Proof
From Fundamental Theorem of Arithmetic, let:

From LCM from Prime Decomposition:


 * $\lcm \set {m, n} = p_1^{\max \set {k_1, l_1} } p_2^{\max \set {k_2, l_2} } \dotsm p_r^{\max \set {k_r, l_r} }$

From GCD from Prime Decomposition:


 * $\gcd \set {m, n} = p_1^{\min \set {k_1, l_1} } p_2^{\min \set {k_2, l_2} } \dotsm p_r^{\min \set {k_r, l_r} }$

From Sum of Maximum and Minimum, for all $i \in \set {1, 2, \ldots, r}$:


 * $\min \set {k_i, l_i} + \max \set {k_i, l_i} = k_i + l_i$

Hence: