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 =  2³
12=  2² · 3    lcm (8, 12, 16) =  2⁴ · 3 = 48
16=  2⁴

2. Calculate lcm (75, 90):
75 =      3 ·   5²
90 = 2 · 3² · 5     lcm (75, 90) = 2 · 3² · 5² = 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