Lowest Common Multiples

Common multiples are multiples that two numbers have in common. These can be useful when working with fractions and ratios.

Example:
What are some common multiples of 2 and 3?
  Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24...
  Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...
Common multiples of 2 and 3 include 6, 12, 18, and 24.

Example:
What are some common multiples of 25 and 30?
  Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325...
  Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330...
Common multiples of 25 and 30 include 150 and 300.

The lowest common multiple or least common multiple is the lowest multiple two numbers have in common.

There are two ways of finding the lowest common multiple of two numbers.

Method 1: Listing Multiples

The first way to find the lowest common multiple is to do what we did above: write out a list of the lowest multiples of each number, and look for the lowest multiple both numbers have in common.

Example:
What is the lowest common multiple of 2 and 3?
  Multiples of 2: 2, 4, 6, 8...
  Multiples of 3: 3, 6, 9...
The lowest common multiple of 2 and 3 is 6.

Example:
What is the lowest common multiple of 25 and 30?
  Multiples of 25: 25, 50, 75, 100, 125, 150, 175...
  Multiples of 30: 30, 60, 90, 120, 150, 180...
The lowest common multiple of 25 and 30 is 150.

Method 2: Factors

The other way to find the lowest common multiple is to list the prime factors for each number. Remove the prime factors both numbers have in common. Multiply one of the numbers by the remaining prime factors of the other number. The result will be the lowest common multiple.

Example:
What is the lowest common multiple of 25 and 30?
  The prime factors of 25 are 5 x 5.
  The prime factors of 30 are 2 x 3 x 5.
  Remove the 5 that 25 and 30 have in common as a prime factor.
  Multiply 25 by the remaining prime factors of 30.
  25 x 2 x 3 = 150.
The lowest common multiple of 25 and 30 is 150.

You'll get the same results no matter which number you work with:

Example:
What is the lowest common multiple of 25 and 30?
  The prime factors of 25 are 5 x 5.
  The prime factors of 30 are 2 x 3 x 5.
  Remove the 5 that 25 and 30 have in common as a prime factor.
  Multiply 30 by the remaining prime factors of 25.
  30 x 5 = 150.
The lowest common multiple of 25 and 30 is 150.

Another Example

Example:
What is the lowest common multiple of 42 and 48?
  The prime factors of 42 are 2 x 3 x 7.
  The prime factors of 48 are 2 x 2 x 2 x 2 x 3.
  Remove the 2 x 3 that 42 and 48 have in common as prime factors.
  Multiply 48 by the remaining prime factors of 42.
  48 x 7 = 336.
The lowest common multiple of 42 and 48 is 336.

What if they have no prime factors in common?

Example:
What is the lowest common multiple of 44 and 45?
  The prime factors of 44 are 2 x 2 x 11.
  The prime factors of 45 are 3 x 3 x 5.
  44 and 45 have no prime factors in common.
Either:
  Multiply 44 by the remaining prime factors of 45.
  44 x 3 x 3 x 5 = 1980.
Or:
  Multiply 45 by the remaining prime factors of 44.
  45 x 2 x 2 x 11 = 1980.
Or:
  44 x 45 = 1980.
The lowest common multiple of 44 and 45 is 1980.

As that last example illustrates, if two numbers have no prime factors in common, the lowest common multiple will be equal to the product of the two numbers.

Treat primes as prime factors

If one number is prime, you can treat it as its own prime factor.

Example:
What is the lowest common multiple of 7 and 30?
  7 is a prime number.
  The prime factors of 30 are 2 x 3 x 5.
  7 and 30 have no prime factors in common.
  7 x 30 = 210.
The lowest common multiple of 7 and 30 is 210.

Example:
What is the lowest common multiple of 2 and 3?
  2 is a prime number.
  3 is a prime number.
  2 and 3 have no prime factors in common.
  2 x 3 = 6.
The lowest common multiple of 2 and 3 is 6.

Example:
What is the lowest common multiple of 3 and 30?
  3 is a prime number.
  The prime factors of 30 are 2 x 3 x 5.
  Remove the 3 that 3 and 30 have in common as a prime factor.
Either:
  Multiply 3 by the remaining prime factors of 30.
  3 x 2 x 5 = 30
Or:
  You would normally multiply 30 by the remaining prime factors of 3, but there are no remaining prime factors.
The lowest common multiple is 30.

Prime results

As you can see from the above, there are two scenarios if at least one number is prime:

• If one number is prime, and the other number's prime factors include that prime number, the lowest common multiple will be equal to the non-prime number.
• If one number is prime, and the other number's prime factors do not include that prime number, the lowest common multiple will be equal to the product of the two numbers.

(The second scenario also includes cases where both numbers are prime.)