**Middle School Number Sense Lesson 26: Finding the Greatest Common Factor**

This concept appeared

**9 times**this year between questions 19 & 27 (inclusive), but most often at

**question # 24**. It is a companion idea to both

**PrimeFactor**(prime factoring) and

**FindLCM**(finding the least common multiple), which I will share a few weeks from now.

**Number Dojo Level: 92**

The greatest common factor (GCF) and the greatest common divisor (GCD) are the same thing.

*GCD is the abbreviation more commonly used on high school tests.*By definition, the

**GCF**of two numbers is the greatest factor that divides into two (or more) numbers without a remainder. In other words, it is the largest factor that two or more numbers have in common.

**How To Solve:**

This is where things get a little magical. Typically GCF is taught in school with prime factoring--where both numbers are factored down to their primes, and then all the common factors are identified and combined. We will solve these instead by

**subtracting**, with maybe a little bit of multiplication thrown in.

The idea behind this concept is that the GCF of any two numbers is also the GCF of the

**difference**between the two numbers, and the original numbers.

*I honestly have no idea why this works--I haven't spent the time to do the painful proofs.*I just know from experience that it works. Here's how I solve GCF questions:

- Subtract the two numbers.
- If the difference is a factor of both numbers, you have your GCF.
- If not, take the smaller of the two original numbers and the difference from step 2. Repeat step 1.

**Example 1: The GCF of 30 and 45 is ___**

- Subtract the two numbers. 45 - 30 =
**15**. - Does 15 go into both 30 & 45 evenly?
**Yes**. You have your GCF.

**Example 2: The GCF of 24 and 18 is ___**

- 24 - 18 =
**6**. - 6 goes into both 24 and 18. You have your GCF.

**Example 3: The GCF of 36 and 54 is ___**

- 54 - 36 =
**18**. - 18 goes into both 36 and 54. You have your GCF.

**Example 4: The GCF of 80 and 32 is ___**

- 80 - 32 =
**48**. - 48 clearly does not go into 32 (or 80).
- Take the smaller original number and the difference; subtract.
**48 - 32 = 16**. - 16 goes into both 32 and 80.
**16**is the GCF.

**Here's where the multiplication comes in handy...**

- If one of the original numbers is more than double the other (or even if it's not), you can multiply the smaller number by 2 (or 3 or 4 or whatever) to get it as close as possible to the larger number.
**THEN**subtract. It'll save you steps.

**Example 4 (Take 2): The GCF of 80 and 32 is ___**

- Multiply 32 by 2 to get 64. 80 - 64 =
**16**. - 16 goes into both 32 and 80. You have your GCF.

**Example 5: The GCF of 18 and 45 is ___**

- Multiply 18 by 2 to get 36. 45 - 36 =
**9**. - 9 goes into both 18 and 45. You have your GCF.

**Example 6: The greatest common factor of 45 and 85 is ___**

- Multiply 45 by 2 to get 90. 90 - 85 =
**5**. - 5 goes into both 45 and 85. You have your GCF.

Example 7: What is the greatest number that divides 12, 18, and 27 without a remainder?

Example 7: What is the greatest number that divides 12, 18, and 27 without a remainder?

*

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

**Note**- 18 - 12 =
**6**. - 6 goes into both 18 and 12.
- 6 x 4 = 24. 27 - 24 =
**3**. - 3 goes into 12, 18, and 27. You have your GCF.

**Example 8: The GCF of 144 and 112 is ___**

- 144 - 112 = 32.
- 32 does not go into 112 or 144.
- 32 x 3 = 96. 112 - 96 =
**16**. - 16 goes into both 144 (9 times) and 112 (7 times).
**16 is the GCF**.

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

findgcf.pdf |

**Up Next for Middle School: CubeRoot**