# Least Common Multiple

How to find the least common multiple (LCM) of the given numbers and monomials: 2 examples and their solutions.

## ExampleLCM of 18 and 30

Finding the least common multiple

is quite similar to

finding the greatest common factor.

Find the prime factorization of 18 and 60.

18 = 2⋅3^{2}

60 = 2^{2}⋅3⋅5

Write 18 = 2⋅3^{2}.

Write 60 = 2^{2}⋅3⋅5 in the next line.

Then find the LCM:

the least common multiple.

Compare the same base powers

and write the greater exponent power

in the LCM.

18 has 2.

60 has 2^{2}.

So write, the greater exponent power, 2^{2}.

18 has 3^{2}.

60 has 3.

So write, the greater exponent power, 3^{2}.

18 doesn't have 5.

60 has 5.

So write, the greater exponent power, 5.

So the LCM of 18 and 60 is

2^{2}⋅3^{2}⋅5.

2^{2} = 4

3^{2} = 9

4⋅5 = 20

20⋅9 = 180

So 180 is the least common multiple of 18 and 60.

## ExampleLCM of 6a^{3}c and 2a^{2}bc^{2}

Write the prime factorizations of the monomials.

Write 6a^{3}c = 2⋅3⋅a^{3}⋅c.

Write 2a^{2}bc^{2} = 2⋅a^{2}⋅b⋅c^{2} in the next line.

Compare the same base powers

and write the greater exponent power

in the LCM.

6a^{3}c has 2.

2a^{2}bc^{2} also has 2.

So write 2.

6a^{3}c has 3.

2a^{2}bc^{2} doesn't have 3.

So write, the greater exponent power, 3.

6a^{3}c has a^{3}.

2a^{2}bc^{2} has a^{2}.

So write, the greater exponent power, a^{3}.

6a^{3}c doesn't have b.

2a^{2}bc^{2} has b.

So write, the greater exponent power, b.

6a^{3}c has c.

2a^{2}bc^{2} has c^{2}.

So write, the greater exponent power, c^{2}.

So the least common multiple of 6a^{3}c and 2a^{2}bc^{2} is

2⋅3⋅a^{3}⋅b⋅c^{2}.

2⋅3 = 6

So 2⋅3⋅a^{3}⋅b⋅c^{2}

= 6a^{3}bc^{2}.

So the least common multiple of 6a^{3}c and 2a^{2}bc^{2} is

6a^{3}bc^{2}.