gcd and lcm via factorization

Statement

Using prime factorization, compute the greatest common divisor and the least common multiple of $24$ and $36$.

Solution

Factor: $24 = 2^3 \cdot 3, \qquad 36 = 2^2 \cdot 3^2.$ The gcd takes each common prime with the smallest exponent: $\gcd = 2^2 \cdot 3 = \boxed{12}.$ The lcm takes each prime with the largest exponent: $\operatorname{lcm} = 2^3 \cdot 3^2 = \boxed{72}.$ Check: $\gcd \cdot \operatorname{lcm} = 12 \cdot 72 = 864 = 24 \cdot 36$.