Let $a=p_1^{x_1}p_2^{x_2}\cdots\cdot p_q^{x_q}$ and $b=p_1^{y_1}p_2^{y_2}\cdots p_r^{x_r}$ http://www.cut-the-knot.org/blue/gcd_fta.shtml, https://math.stackexchange.com/a/349867/85100 say —
Since gcd(a,b) is the largest common divisor of a and b and is divisible by any other common divisor of the two,
GCD$(a,b)=p_1^{\min(x_1,y_1)}p_2^{\min (x_2,y_2)} \cdots p_q^{\min (x_q,y_q)}$
LCM$(a,b)= p_1^{\max(x_1,y_1)}p_2^{\max(x_2,y_2)}\cdots p_r^{\max (x_r,y_r)}$
I understand GCD(a,b) has to divide both $a,b$. Therefore the exponent of any prime factor $p_i$ in GCD has to be in both $a,b$ — therefore $p_i^{\min(x_i, y_i)}$. But I'm muddled and anxious. Why does $\min$ appear in GREATEST common divisor? Why does $\max$ appear in LOWEST common multiple?