LCM of 84 and 120

Statement

Compute the least common multiple of $84$ and $120$.

Solution

Factor the two numbers: $84 = 2^2\cdot 3\cdot 7, \qquad 120 = 2^3\cdot 3\cdot 5.$ The LCM takes each prime factor with the highest exponent: $\operatorname{lcm} = 2^3\cdot 3\cdot 5\cdot 7 = \boxed{840}.$ (For comparison, the GCD takes the lower exponents: $2^2\cdot 3 = 12$.)