Least Common Multiple (LCM) | Elementary

The least common multiple (LCM) is a multiple that is shared by two or more numbers. Numbers can have more than one multiple in common, but the LCM designates the smallest one. The LCM is never |0|.

To find the LCM using the multiples method, I have to follow these steps.

1. I write a list of the first few multiples for each of the numbers.

2. I determine the multiples in common in the set of numbers.

3. I identify the smallest of these multiples.

What is the LCM of |10,| |12,| and |15?|

1. I write a list of the first multiples for each of the numbers.

2. I determine the multiples in common in the set of numbers.

3. I identify the smallest of these multiples.

The LCM of |10,| |12,| and |15| is |60.|

To find the LCM using prime factorization, I have to follow these steps.

1. I make a factor tree for each number.

2. I find the prime factors of each number.

3. I identify prime factors shared by at least two numbers and write them down.

4. I calculate the LCM by multiplying together the shared prime factors and the remaining prime factors (which are alone).

What is the LCM of |10,| |12,| and |15?|

I make a factor tree for each number. Note: To learn how to make a factor tree, see this concept sheet: The Factor Tree.
I find the prime factors of each number. Note: To learn what a prime factor is, see this concept sheet: Prime Factors and Prime Factorization.
I identify the prime factors shared by at least two numbers and write them down. The shared prime factors are |2,| |5| and |3.|
I calculate the LCM by multiplying together the common prime factors and the remaining prime factors (which are alone).

The LCM of |10,| |12,| and |15| is |60.|

To find the LCM using a prime factor table, I have fo follow these steps.

1. I draw a table and label the first column Prime Factors. The labels of the other columns will correspond to the numbers whose LCM I need to find.

2. I try to divide each of the numbers by |2.| If division is possible, I write |2| in the Prime Factors column and the result of the division under each number.

3. I repeat until I no longer get a number that divides by |2.|

4. Then, I try to divide the numbers by |3,| then by |4,| |5,| |6,| and so on until I get |1| in each column.

5. I calculate the LCM by multiplying the prime factors in the first column.

What is the LCM of |10,| |12,| and |15?|

I draw a table and label the first column Prime factors. The labels of the other columns correspond to the numbers whose LCM I need to find.
I try to divide the numbers by |2.| If a division is possible, I write |2| in the Prime divisors column and the result of the division under each number.  |10\div2 = 5| |12\div2 = 6|
I repeat until I no longer have a number that is divisible by |2.| |6\div2 = 3|
Next, I try to divide the last number entered into each column except the Prime Factors. I start by dividing by |3,| then by |4,| |5,| |6,| and so on until each column has |1.| |3\div3 = 1| |15\div3 = 5| |5\div5 = 1|
I calculate the LCM by multiplying all of the prime factors in the first column.	2 × 2 × 3 × 5 = 60

The LCM of |10,| |12,| and |15| is |60.|