Math Problem Solver

Method 1:

The least common multiple of a set of integers is the smallest positive integer divisible by all integers in the set.

The prime factorization of $84$ is $2^{2} \times 3 \times 7$.

The prime factorization of $108$ is $2^{2} \times 3^{3}$.

Find the least common multiple by multiplying the prime number powers together, including only the greatest prime number power of each common prime.

$\text{lcm}(84, 108) = 2^{2} \times 3^{3} \times 7 = 756.$

Method 2:

The least common multiple of a set of integers is the smallest positive integer divisible by all integers in the set.

The lcm of two numbers is related to the greatest common denominator of those same two numbers by the following relation: $\text{lcm}(a, b) = \frac{a \times b}{\text{gcd}(a, b)}$.
Therefore:

$\text{gcd}(84, 108) = 12$

$\text{lcm}(84, 108) = \frac{84 \times 108}{12} = \frac{9072}{12} = 756$