# Prime Numbers

 Definition: A natural number is a prime number if the only factors of it are 1 and itself. That is, no other number divides into it evenly (without remainder).   The prime numbers less than 100 are: 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 Definition: A natural number that is not a prime number is called a composite number. Rule (4th-6th grade): To determine whether a number <50 is prime, we test whether any prime number less than half of it is a factor. Rule (7th grade up): To determine whether a number is prime, we test whether any prime number less than its square root is a factor. Demonstration - Is it a prime? == Determine whether the number 43 is prime. (4th-6th Grade) 1. Half of 43 is 22. The prime numbers smaller than 22 are 2, 3, 5, 7, 11, 13, 17 and 19. 2. Test these 8 numbers.    a. 43 is not even, so 2 is not a factor.    b. 43 / 3 is 12+ so 3 is not a factor or say the sum of the digits of 43 is 7, which is not divisible by 3...    c. 43 doesn't end with a 0 or a 5 so 5 is not a factor.    d. 43 / 11 = 3+ so 11 is not a factor.    e. 43 / 13 = 3+ so 13 is not a factor.    f. 43 / 17 = 2+ so 17 is not a factor.    g. 43 / 19 = 2+ so 19 is not a factor. 3. Conclusion: 43 is a prime number. == Determine whether the number 91 is prime. (7th Grade & up) 1. The closest perfect square bigger than 91 is 100. The square root of 100 is 10. The prime numbers smaller than 10 are 2, 3, 5 and 7. 2. Test these 4 numbers.    a. 91 is not even, so 2 is not a factor.    b. 91 / 3 is 30+ so 3 is not a factor. (The sum of the digits of 91 is 10, which is not divisible by 3...)    c. 91 doesn't end with a 0 or a 5 so 5 is not a factor.    d. 91 / 7 = 13 so 7 is a factor. 3. Conclusion: 91 is not a prime number; 91 is a composite number. == Determine whether the number 149 is prime. (7th Grade & up) 1. The closest perfect square bigger than 149 is 169. The square root of 169 is 13. The prime numbers smaller than 13 are 2, 3, 5, 7 and 11 2. Test these 5 numbers.    a. 149 is not even, so 2 is not a factor.    b. 149 / 3 = 49+ so 3 is not a factor. (The sum of the digits of 149 is 14, which is not divisible by 3...)    c. 149 doesn't end with a 0 or a 5 so 5 is not a factor.    d. 149 / 7 = 21+ so 7 is not a factor.    e. 149 / 11 = 13+ so 11 is not a factor. 3. Conclusion: 149 is a prime number.
Definition: Prime factorization or prime decomposition is to write a composite number as a product of its prime factors.

Demonstration - Prime Factorisation (7th Grade & up)

 Step 1: Make a factorisation tree:         a. Divide the number by all the prime numbers less than "the square root of number" until we get to a prime.         b. If none of these primes divide, the number itself is prime. Step 2: Form the prime factorisation from this tree. Step 3: The factors are: 1, the number itself and all combinations of products of the prime factorisation. Step 4: Check the factor pairs to make sure nothing was missed.
1. Find the prime factorisation and factors of 18.   Mathcast
Answer: 18 = 2 \times 3 \times 3 = 2 \cdot 3^2 . Factors are: 1, 2, 3, 6. 9, 18
2. Find the prime factorisation and factors of 308.  Mathcast
Answer: 308 = 2 \times 2 \times 7 \times 11 = 2^2 \cdot 7 \cdot 11 . Factors are: 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308
3. Find the prime factorisation and factors of 91.   Mathcast Answer: 91 = 7 \times 13 . Factors are: 1, 7, 13, 91
4. Find the prime factorisation and factors of 201.  Mathcast Answer: 201 = 3 \times 67 . Factors are: 1, 3, 67, 201
5. Find the prime factorisation and factors of 191.  Mathcast Answer: 191 is a prime number so no prime decomposition. Factors are: 1, 191.
6. Find the prime factorisation 1120. Mathcast Answer: 1020=2^5 \cdot 5 \cdot 7 . There are 24 factors!

 Test your factorization skills (7th grade & up) Install the ExamView Player (freeware) from Get EV Player Go for it! Prime Factorisation Quiz
Standards for 4th-6th grade: CA 4.NS.4.0, CA 5.NS.1.4, ACT Number Sense 20-23
Standards for 7th grade & up: ACT Number Sense 24-28

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