**Middle School Number Sense Lesson 33: Prime Factors**

This concept appeared

**12 times**this year with a median spot of

**question # 32**. You already had a little taste of prime factors a couple of weeks ago when I introduced

**NumPosIntDiv**as a high school lesson. This one isn't nearly as complicated, but it shows up about as often.

**Number Dojo Level: 175**

On number sense tests, you will see a few variations of this concept. In descending order of frequency, they are:

- (about 58% of the time): The number of distinct prime divisors of 30 is ___
- (about 30% of the time): The largest prime factor of 510 is ___
- (about 10% of the time): Find the sum of the distinct prime divisors of 84. ___
- (about 2% of the time): The smallest prime divisor of 189 is ___

**How to Solve:**

There is no known formula or algorithm that generates a list of prime factors quickly. A number sense student learns to recognize prime numbers and divisibility rules quickly. I will demonstrate how I prime factor any number:

- See if your original number divides by 2, then by 3, then by 5, then by 7, etc., until you have tried every prime number less than or equal to half your original number.
- As you identify factors, remove them from your process to make the math easier.
- Keep track of each prime factor along the way.

**Example 1: The number of distinct prime divisors of 30 is ___**

- 30 is even, so it is divisible by
**2**. 30 ÷ 2 = 15. - 15 is obviously divisible by
**5**. 15 ÷ 5 =**3**, which is prime. - Your prime divisors are
**2**,**3**, and**5**. There are**3**distinct prime divisors of 30.

**Example 2: The largest prime factor of 510 is ___**

- 510 is obviously divisible by 10, which is
**2**x**5**. 510 ÷ 10 =**51**. - 51 is divisible by
**3**. 51 ÷ 3 =**17**, which is prime. - Your prime factors are
**2**,**3**,**5**, and**17**. The largest one is**17**, which is your answer.

**Example 3: Find the sum of the distinct prime divisors of 84. ___**

- 84 is even, so it is divisible by
**2**. 84 ÷ 2 = 42. 42 is also even; 42 ÷ 2 = 21. - 21 is divisible by
**3**. 21 ÷ 3 =**7**, which is prime. - Your prime factors are
**2**,**3**, and**7**. 2 + 3 + 7 =**12**, which is your answer.

**Example 4: The smallest prime divisor of 189 is ___**

- 189 is not even, so it is not divisible by 2.
- Add up the digits of 189. 1 + 8 + 9 = 18, which is divisible by 3. So 189 is divisible by 3.
- You can stop there, because you have already found your smallest prime divisor:
**3**.

**Example 5: 2016 has how many distinct prime divisors?**

- 2016 is even, so it is divisible by
**2**. 2016 ÷ 2 = 1008. 1008 is also even, so keep going. 1008 ÷ 2 = 504. 504 ÷ 2 = 252. 252 ÷ 2 = 126. 126 ÷ 2 = 63. - 63 is divisible by
**3**. 63 ÷ 3 = 21. 21 ÷ 3 =**7**, which is prime. - Your prime divisors are
**2**,**3**, and**7**. Your answer is**3**.

**Example 6: The largest prime factor of 780 is ___**

- 780 is obviously divisible by 10, which is
**2**x**5**. 780 ÷ 10 =**78**. - 78 is even, so it is divisible by 2. 78 ÷ 2 =
**39**. - 39 is divisible by
**3**. 39 ÷ 3 =**13**, which is prime. - Your prime factors are
**2**,**3**,**5**, and**13**. The largest is**13**, which is your answer.

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

primefactor.pdf |

**Up Next for Middle School: MultMixF+1**