**Middle School Number Sense Lesson 30: Prime Numbers**

I'm a big fan of prime numbers. After all these years of participating in number sense, it's hard for me to see a prime number (

**anywhere**) and NOT think "that's a prime number." I've learned not to say it out loud each time I think it though--because I value friends.

Questions about prime numbers show up on most tests; this year they appeared

**13 times**near

**question # 27**. (That's in addition to questions about twin primes, prime factors, and relative primes--which we'll discuss later).

**Number Dojo Level: 71**

A prime number, by definition, is a natural number greater than 1 that has no positive divisors other than 1 and itself. (

*Thank you, Wikipedia!*) I personally prefer to define a prime number as having

**exactly two positive factors**(1 and itself). Most people I talk to don't know that prime has an antonym:

**composite**(a number >1 with more than 2 positive factors).

There is no simple way to identify prime numbers. For number sense purposes, they must either be memorized or tested for

**primality**.

It is easy to memorize the first 8 prime numbers; I recommend doing so. There are

**exactly 25 primes less than 100**(I do not recommend memorizing the list):

**2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97**

**in their prime**.

Another fun fact is that there are

**168 prime numbers less than 1000**. Why is that fun? Let me answer that by asking you this: how many hours are in a week? Okay then!

**Testing a Number for Primality**

- Find the largest prime number whose square is
**less than**the number you are investigating. - Divide the number by all the primes
**up to**the prime you just identified. - If none of these primes divides evenly into your number, then
**it is prime**.

**Example 1: Which number is prime: 37 or 39?**

- We will test 37. The largest prime whose square is < 37 is
**5**. (7 x 7 = 49). - 37 ÷ 2 has a remainder of 1. 37 ÷ 3 has a remainder of 1. 37 ÷ 5 has a remainder of 2.
- Since none of these primes divides evenly,
**37 is prime**. There is no need to check 39 (for purposes of this question).

**Example 2: Which is prime: 61, 63, or 65?**

- Let's test 61. The largest prime whose square is < 61 is
**7**. (11 x 11 = 121). - 61 ÷ 2 has R 1. 61 ÷ 3 has R 1. 61 ÷ 5 has R 1. 61 ÷ 7 has R 5.
- Since none of these primes divides evenly,
**61 is prime**. There is no need to check the other two.

**Example 3: What is the largest prime number less than 100?**

- Working backwards: 99 is divisible by 3.
- 98 is divisible by 2.
- 97 may be prime...let's test it. The largest prime whose square is < 97 is
**7**. (11 x 11 = 121). - 97 ÷ 2 has R 1. 97 ÷ 3 has R 1. 97 ÷ 5 has R 2. 97 ÷ 7 has R 6.
- Since none of these primes divides evenly,
**97 is prime**.

Example 4: What is the smallest prime number greater than 62?

Example 4: What is the smallest prime number greater than 62?

- Working forward: 63 is divisible by 3.
- 64 is divisible by 2.
- 65 is divisible by 5.
- 66 is divisible by 2.
- 67...let's test it. The largest prime whose square is < 67 is
**7**. (11 x 11 = 121). - 67 ÷ 2 has R 1. 67 ÷ 3 has R 1. 67 ÷ 5 has R 2. 67 ÷ 7 has R 4.
- Since none of these primes divides evenly,
**67 is prime**.

**Example 5: What is the sum of the first 6 prime numbers?**

- Memorizing (or quickly testing) gives you
**2 + 3 + 5 + 7 + 11 + 13**. - The answer is
**41**. Also a prime number :).

**Example 6: What is the product of the first 4 prime numbers?**

- Memorizing (or quickly testing) gives you
**2 x 3 x 5 x 7**. - The answer is
**210**.

**Example 7: How many prime numbers are between 20 and 30?**

- Memorizing (or quickly testing) gives you
**23 and 29**. - There are
**2**.

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

prime.pdf |

**Up Next for Middle School: Mult111**