What is the least common multiple of 108 and 480?

Answer 1

The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both of these numbers. To find the LCM of 108 and 480, we can use different methods, such as prime factorization, listing multiples, or using the greatest common divisor (GCD). Here, I will explain the LCM using the prime factorization method:

Prime Factorization:

First, find the prime factors of both numbers:

• 108 = 2^2 × 3^3
• 480 = 2^5 × 3 × 5

Then, for each Prime number, take the highest power found in both factorizations:

• For 2, the highest power is 5 (from 480).
• For 3, the highest power is 3 (from 108).
• For 5, the highest power is 1 (from 480).

Multiply these highest powers of all prime factors to get the LCM:

• LCM = 2^5 × 3^3 × 5 = 32 × 27 × 5 = 4320

The least common multiple of 108 and 480, therefore, is 4320, indicating the smallest number into which both 108 and 480 can divide evenly. This concept is particularly useful in solving problems involving fractions, multiples, and periodic events in mathematics.

Answer 2

108 = 2 x 2 x 3 x 3 x 3; 480 = 2 x 2 x 2 x 2 x 2 x 3 x 5.

Common factors 2 x 2 x 3 so LCM = 9 x 480 = 4320 (also = product/HCF)