# Lowest common multiple (lcm)

The **lowest (least) common multiple** is the smallest number that is a common multiple of two or more numbers.

Examples:

1. Calculate the lowest common multiple of 12 and 15:

Common multiples(12, 15)={60, 120,…} lowest common multiple, lcm(12, 15) = 60

2. Calculate the lowest common multiple of 8, 12 and 16:

Common multiples(8,12, 16)={48, 96,…} lowest common multiple, lcm(8,12, 16) = 48

This way to find the lcm is not optimum to calculate with high numbers. Then we use the ''*optimum way*'':

- Decompose all numbers in prime factors.

- Multiply all the prime factors (common and not common) to the maximum index.

Examples:

1. Calculate the lowest common multiple of 8, 12 and 16:

8 = 2^{3}

12= 2^{2} · 3 lcm (8, 12, 16) = 2^{4} · 3 = 48

16= 2^{4}

2. Calculate lcm (75, 90):

75 = 3 · 5^{2}

90 = 2 · 3^{2} · 5 lcm (75, 90) = 2 · 3^{2} · 5^{2} = 450

**Exercise**.- Calculate the lcm of:

a) 16 and 18

b) 15 and 20

c) 8, 12 and 18

Solution: a) 144; b) 60; c) 72

Licensed under the Creative Commons Attribution Non-commercial Share Alike 3.0 License