Prime and Composite Numbers Factors What is a prime number? Is this number prime? Teaching Prime Numbers Elementary school children can learn prime numbers, and the process can become an exciting challenge. As teachers, we can do much to make this a good learning experience for our students. We begin simply, by reviewing multiplication Multiplication 2 x 3 = ____ Yes, we start with an easy one. We know that 2 x 3 = 6 We remind our students that: 2 and 3 are factors 6 is the product Showing Factors Here are 6 blocks We can arrange them to make a rectangle That is 3 blocks long and 2 blocks wide This shows that 2 and 3 are factors. Finding the factors of 6 We have already shown that 2 and 3 are factors of 6. Are there any others? We know that 1 x 6 = 6 So, 1 and 6 are also factors of 6. That means 1, 2, 3, and 6 are factors. Are there any more? Other Factors of 6 Here are 6 blocks We can arrange them to make a rectangle That is 6 blocks long and 1 block wide This shows that 1 and 6 are also factors of 6. That means 1, 2, 3, and 6 are factors. Are there any more? The only other numbers we can try are 4 and 5. Lets see if we can make rectangles. Testing other factors Can we make a rectangle with 6 blocks that is 4 blocks long? No, we cant. We would need two more blocks to make a rectangle. The Factors of 6 We can make a rectangle that is 2 x 3 blocks. That means 2 and 3 are factors of 6. We can make a rectangle that is 1 x 6 blocks. That means 1 and 6 are factors of 6. Those are the only rectangles with 6 blocks. We know that 1, 2, 3, and 6 are the only factors of 6. Making a List The teacher helps students make a list of every factor for every number up to 10. The factors of 6 have already been determined. Beginning the Factor List , 2, 3, A Factor List 11 21, 2 31, 3 41, 2, 4 51, 5 61, 2, 3, 6 71, 7 81, 2, 4, 8 91, 3, 9 101, 2, 5, 10 What Do We Notice? The only number with only one factor is 1. Every number greater than 1 has at least two factors, 1 and the number itself: these are the identity factors, 1 x n = n. If there are two factors, and only two factors, then the number is prime. If there are more than two factors, the number is composite. Finding the Primes In looking at the factor list, we see that 2, 3, 5, and 7 are prime numbers. For composite numbers (4, 6, 8, 9, and 10) the smallest factor (other than 1) is always a prime number. It is called the Smallest Prime Factor (SPF). Prime and Composite Numbers 1Neither 2Prime 3Prime 4Composite (2) 5Prime 6Composite (2) 7Prime 8Composite (2) 9Composite (3) 10Composite (2) The Next Steps We could continue this process of finding all the factors of a number, and noticing whether or not there are exactly two factors. This is a good thing for children to do up to 20 or 25. However There is a much easier method... The Sieve of Eratosthenes Eratosthenes lived over 2000 years ago. He invented a method for finding prime numbers that is still used today. The method is brilliant and very simple. Start with a 100-chart. Cross out 1. Blot out the number 1 Look at the next number, 2. It has no color. So, we circle 2. Then we cross out every multiple of 2. Circle 2 and color its multiples Look at the next number, 3. It has no color. So, we circle 3. Then we cross out every multiple of 3. Circle 3 and color its multiples Look at the next number, 4. It DOES HAVE color. So, we ignore it. Look at the next number, 5. It has no color. So, we circle 5. Then we cross out every multiple of 5. Circle 5 and color its multiples Look at the next number, 6. It DOES HAVE color. So, we ignore it. Look at the next number, 7. It has no color. So, we circle 7. Then we cross out every multiple of 7. Circle 7 and color its multiples The Sieve is Complete We notice that 8, 9, and 10 have already been crossed out. We circle 11 and begin to cross out its multiples. We notice that every multiple of 11 has already been crossed out! We have found all the prime numbers! Lets Show it in Color It looks better in color. We can also see more mathematical ideas. A Colorful Sieve What Tests Do We Need? So we do need to test 2, 3, 5, and 7. Do we need to test any other numbers? In particular, do we need to test whether a number is divisible by 11? 11, is a prime number. The other multiples are: 22, 33, 44, 55, 66, 77, 88, and , 44, 66, and 88 are also divisible by 2; 33 and 99 are divisible by 3; 55 is divisible by 5; 77 is divisible by 7. The Multiples of 11 11, is a prime number. The other multiples are: 22, 33, 44, 55, 66, 77, 88, and , 44, 66, and 88 are also divisible by 2 33 and 99 are divisible by 3 55 is divisible by 5 77 is divisible by 7 Divisibility Tests After a thorough discussion of the Sieve of Eratosthenes, students will know that 2, 3, 5, and 7 are prime numbers, and that 4, 6, 8, 9, and 10 are composite numbers. Furthermore, only four divisibility tests need to be performed for any number in the 100-chart: 2, 3, 5, and 7. If a number is divisible by any of those four factors, the number is composite. If it is not divisible, then it must be prime. The Easy Divisibility Tests A number divisible by: 2 if it is even 5 if if ends in 0 or 5 The Divisibility Test for 3 A number is divisible by 3: if the sum of the digits is 3, 6, or 9. For 87, we add the digits: = 15. Then we continue the process: = 6, and we see that 87 is divisible by 3. For 71, we add the digits: = 8. and we see that 71 is NOT divisible by 3. The Multiples of 7 There is no easy divisibility test for 7. It is useful to recall the multiples of 7: The Divisibility Test for 7 But we dont have to remember all. After we test for 2, 3, and 5; Only three multiples of 7 are left: Divisibility Tests: Summary 2 -> Even 3 -> Digit sum is 3, 6, or 9 5 -> Ends in 0 or 5 7 -> 49, 77, 91 All other multiples of 7 have been eliminated Example 82 is even This means that is has a factor of 2. This means there are more than two factors. [We know that 1 and 82 are factors.] Therefore, 82 is a composite number. Example 57 has a digit sum: = = 3 This means that is has a factor of 3. This means there are more than two factors. Therefore, 57 is a composite number. Example Test 91 Is odd, so 2 is not a factor Digit sum = 10, so 3 is not a factor Ends in 1, so 5 is not a factor, But 91 is divisible by 7 [7 x 13 = 91] Therefore, 91 is a composite number. Example Test 71 2 Odd 3 Digit sum = Ends in 1 7 Not divisible by 7 71 is not divisible by 2, 3, 5 or 7; it is prime Summary The prime numbers less than 10 are 2, 3, 5, or 7 For any number from 10 to 100 If it is divisible by 2, 3, 5, or 7 then it is composite. Otherwise the number is prime.