Lesson 6: Least Common Multiple

Property

At the other side of finding common factors of two numbers is that of finding common multiples of two numbers.
There will be a least common multiple and that is designated the LCM.
The LCM is the smallest number that contains all the prime factors of both numbers, taking the larger of the multiples for any shared prime factors.

Examples

• Find the LCM of 8 and 10. Multiples of 8: {8, 16, 24, 32, 40,...}. Multiples of 10: {10, 20, 30, 40,...}. The LCM is 40.
• Find the LCM of 18 and 24 using prime factors. $18 = 2 \cdot 3 \cdot 3$. $24 = 2 \cdot 2 \cdot 2 \cdot 3$. The LCM needs the highest power of each prime: $2^3 \cdot 3^2 = 8 \cdot 9 = 72$.
• Two gears with 12 and 16 teeth mesh. The LCM of 12 and 16 is 48. The starting teeth will align again after the first gear makes 4 turns and the second makes 3 turns.

Explanation

The LCM is the smallest number that two or more numbers can both multiply into. It's the first finish line they both cross in a race of multiples. This is the key to adding and subtracting fractions.

Property

To find the LCM by listing multiples:
(1) List the first several multiples of each number;
(2) Identify multiples that appear in all lists;
(3) Select the smallest common multiple as the LCM.

Examples