**High School Number Sense Lesson 32: Finding the Least Common Multiple**

This concept is an extension of the

**FindGCF**skill, which you should review if you haven't mastered it. Finding the LCM was asked

**6 times**this year--as early as question # 11 and as late as question # 32, on average at

**question # 21**.

**Number Dojo Level: 104**

The

**least common multiple**of two numbers is the smallest positive integer that can be divided evenly by both numbers without a remainder. In other words, the LCM is the smallest number that has both of the others as factors. For example, the LCM of 12 and 16 is 48--because 48 is the smallest positive integer divisible by both 12 and 16. Now let's talk about how we

*find the LCM*:

**The Old Way**

If you're not taking a number sense test, it may be quick and easy to list the multiples (in order) of each number and then identify the first one the numbers have in common. For example, the multiples of 12 are 12, 24, 36,

**48**, 60, 72.... The multiples of 16 are 16, 32,

**48**... We don't have to go any further because both lists have

**48**in them.

But on number sense tests, we don't have the luxury of using scratch paper...so how do we do this in our heads?

**The Mental Way**

- Find the GCF of the two numbers (review
**FindGCF**if necessary). - Divide the product of the two numbers by their GCF.
**This is easiest if you divide one of the numbers by the GCF first, and then multiply that quotient by the other number.**You may or may not know this little gem, but**the product of the GCF and LCM of any two numbers is equal to the product of the two numbers**.

**Example 1: Find the LCM of 20 and 24.**

- Find the GCF first:
**4**. - Divide one of the numbers by the GCF & multiply by the other number. 20
**5**. 5 x 24 =**120**, which is your LCM. (You could have divided 24 by 4 to get 6, and then multiplied by 20 to get 120. It doesn't matter which number you divide by the GCF).

**Example 2: Find the LCM of 15 and 27.**

- Find the GCF first:
**3**. - Divide one number by the GCF & multiply by the other number. 15 ÷ 3 =
**5**. 5 x 27 =**135**, which is your LCM.

**Example 3: The LCM of 54 and 24 is ___**

- Find the GCF first:
**6**. - Divide one number by the GCF & multiply by the other number. 24 ÷ 6 =
**4**. 4 x 54 =**216**. This is your answer.

**Example 4: The LCM of 57 and 95 is ___**

- Find the GCF first:
**19**. - Divide one number by the GCF & multiply by the other number. 57 ÷ 19 =
**3**. 3 x 95 =**285**. This is your LCM.

**Example 5: The LCM of 24, 32, and 36 is ___**

*

__: Here we are asked to find the LCM of 3 numbers. Pick any 2 numbers, find their LCM, and then find the LCM of that number and the third original number.__

**Note**- Find the GCF of 24 and 32 first:
**8**. - Divide one of those numbers by their GCF & multiply by the other number. 24 ÷ 8 =
**3**. 3 x 32 =**96**, so the LCM of 24 and 32 is 96. - Now find the GCF of 96 and 36:
**12**. - Divide one of those numbers by their GCF & multiply by the other number. 36 ÷ 12 =
**3**. 3 x 96 =**288**. You have found the LCM of 24, 32, and 36.

**Example 6: The LCM of 20, 28, and 35 is ___**

- Find the GCF of 20 and 28 first:
**4**. - Divide one of those numbers by their GCF & multiply by the other number. 20 ÷ 4 =
**5**. 5 x 28 =**140**. - Now find the GCF of 140 and 35:
**35**. - Divide one of those numbers by their GCF & multiply by the other number. 35 ÷ 35 =
**1**. 1 x 140 =**140**. You have found the LCM of 20, 28, and 35.

__: In step 3, once you realize the GCF of two numbers (140 & 35) is one of them (35), you know that their LCM automatically has to be the other number (140).__

**Note****Here's a free worksheet to help you practice FindLCM:**

findlcm.pdf |

**Up Next for High School: SubstDirect**