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 (4th6th 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. (4th6th 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)
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)

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