Find the least common multiple of 8, 12 and 18.

How we read your problem

Find the least common multiple of the numbers.

Step-by-step solution

1. 1

   Prime factorization of each number

   $$8 = 2^{3},\quad 12 = 2^{2} \times 3,\quad 18 = 2 \times 3^{2}$$
2. 2

   Use the GCF to combine 8 and 12

   $$\operatorname{lcm}(8, 12) = \frac{8 \times 12}{\gcd(8, 12)} = \frac{96}{4} = 24$$
3. 3

   Use the GCF to combine 24 and 18

   $$\operatorname{lcm}(24, 18) = \frac{24 \times 18}{\gcd(24, 18)} = \frac{432}{6} = 72$$
4. 4

   Confirm with prime factors

   Take every prime that appears, each with its largest exponent.

   $$2^{3} \times 3^{2} = 72$$

Formulas used

• LCM from GCF

  $$\operatorname{lcm}(a, b) = \frac{|a \cdot b|}{\gcd(a, b)}$$

72 is the smallest positive number that every input divides exactly.

Verification

• 72 ÷ 8 = 9
• 72 ÷ 12 = 6
• 72 ÷ 18 = 4