H.4 Identify prime and composite numbers
What are prime numbers?
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. This means the only way to multiply two whole numbers to get that number is 1 times itself.
- 2 is a prime number. Its only factors are 1 and 2: 1 × 2 = 2.
- 3 is a prime number. Its only factors are 1 and 3: 1 × 3 = 3.
- 5 is a prime number. Its only factors are 1 and 5: 1 × 5 = 5.
- 7 is a prime number. Its only factors are 1 and 7: 1 × 7 = 7.
- 11 is a prime number. Its only factors are 1 and 11: 1 × 11 = 11.
The number 1 is not a prime number because it has only one factor (1). Remember: prime numbers must have exactly two different factors.
What are composite numbers?
A composite number is a whole number greater than 1 that has more than two factors. Composite numbers can be divided evenly by numbers other than 1 and themselves.
- 4 is a composite number. Its factors are 1, 2, and 4: 1 × 4 = 4 and 2 × 2 = 4.
- 6 is a composite number. Its factors are 1, 2, 3, and 6: 1 × 6 = 6 and 2 × 3 = 6.
- 8 is a composite number. Its factors are 1, 2, 4, and 8: 1 × 8 = 8 and 2 × 4 = 8.
- 9 is a composite number. Its factors are 1, 3, and 9: 1 × 9 = 9 and 3 × 3 = 9.
- 10 is a composite number. Its factors are 1, 2, 5, and 10: 1 × 10 = 10 and 2 × 5 = 10.
All composite numbers can be broken down into smaller factors. Some composite numbers, like 4 and 9, are called perfect squares because they are made by multiplying a number by itself.
How do factors help identify prime and composite numbers?
Factors are whole numbers that multiply together to make another number. Finding all factor pairs for a number helps you decide if it is prime or composite.
- Step 1: Think of multiplication facts for 12: 1 × 12, 2 × 6, 3 × 4.
- Step 2: List all factors: 1, 2, 3, 4, 6, 12.
- Step 3: Count the factors. 12 has six factors, which is more than two.
- Conclusion: 12 is a composite number.
- Step 1: Think of multiplication facts for 13: 1 × 13 is the only way.
- Step 2: List all factors: 1 and 13.
- Step 3: Count the factors. 13 has exactly two factors.
- Conclusion: 13 is a prime number.
Always check all numbers up to the square root of the number you are testing. If you find a factor other than 1 and itself, the number is composite.
What about 0 and 1?
The numbers 0 and 1 are special cases. They are neither prime nor composite. This is an important rule to remember when identifying number types.
- Number 1: It has only one factor (1). Prime numbers need exactly two factors. Composite numbers need more than two. So 1 fits neither category.
- Number 0: Zero has an infinite number of factors because any number times 0 equals 0. It does not fit the definition of prime or composite, which only applies to numbers greater than 1.
When someone asks, "Is 1 prime?" the correct answer is no. And 0 is not composite. They are in a group of their own.
Prime numbers from 1 to 100
Here is the complete list of prime numbers between 1 and 100. Memorizing these can help you quickly identify primes in this range.
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
Notice that 2 is the only even prime number. Every other even number is composite because it has 2 as a factor.
Composite numbers from 1 to 100
Most numbers from 1 to 100 are composite. Here are some examples to recognize patterns.
- Even numbers greater than 2: 4, 6, 8, 10, 12, 14 ... 100 (all even numbers except 2 are composite).
- Odd composite numbers: 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99.
- Numbers like 21 (3 × 7), 25 (5 × 5), and 49 (7 × 7) are composite.
If a number is not prime and greater than 1, it must be composite. There are 74 composite numbers between 1 and 100 (including 1, which is neither, so actually 74 composites from 4 to 100).
Using divisibility rules to test for primes and composites
Divisibility rules are shortcuts that help you determine if one number divides another evenly. They are useful for quickly finding factors.
- Divisible by 2: The number ends in 0, 2, 4, 6, or 8. (Example: 34 ends in 4, so it is divisible by 2.)
- Divisible by 3: The sum of the digits is divisible by 3. (Example: 57 → 5 + 7 = 12, and 12 is divisible by 3, so 57 is divisible by 3.)
- Divisible by 5: The number ends in 0 or 5. (Example: 85 ends in 5, so it is divisible by 5.)
- Divisible by 10: The number ends in 0. (Example: 80 ends in 0, so it is divisible by 10.)
- Does it end in 0,2,4,6,8? No, so not divisible by 2.
- Sum of digits: 9 + 1 = 10. 10 is not divisible by 3, so not divisible by 3.
- Ends in 5 or 0? No, so not divisible by 5.
- But check 7: 7 × 13 = 91, so it has factors 7 and 13. Therefore 91 is composite.
Divisibility rules help you eliminate possibilities, but sometimes you still need to test dividing by prime numbers like 7, 11, or 13.
How to tell the difference quickly
When you look at a number, ask yourself these three questions to decide if it is prime or composite.
- Is the number less than 2? If yes, it is neither prime nor composite.
- Can you find any factor other than 1 and itself? If yes, it is composite.
- If the only factors are 1 and itself, it is prime.
- 23: Check small primes: not divisible by 2,3,5 (2+3=5 not multiple of 3). Try 7: 7×3=21, 7×4=28, so no. So 23 is prime.
- 27: Sum digits 2+7=9, divisible by 3, so 27 has factor 3. 27 ÷ 3 = 9. So 27 is composite.
- 29: Not divisible by 2,3,5 (2+9=11 not multiple of 3). Check 7: 7×4=28, 7×5=35, so no. So 29 is prime.
You only need to test divisibility by prime numbers up to the square root of the number. For numbers under 100, test primes 2, 3, 5, and 7. For numbers under 121, test up to 11.
Why do we learn about prime and composite numbers?
Prime numbers are the building blocks of all numbers. Every composite number can be written as a product of prime numbers. This is called prime factorization.
- Computer security uses prime numbers to keep information safe online.
- Understanding factors helps with fractions, simplifying, and finding common denominators.
- Many puzzles and patterns in math rely on prime numbers.
- Take 30. Break it into factors: 30 = 3 × 10. Then 10 = 2 × 5. So 30 = 2 × 3 × 5.
- 2, 3, and 5 are prime numbers. So 2 × 3 × 5 is the prime factorization of 30.
Every composite number has one unique set of prime factors (order doesn't matter). This is called the Fundamental Theorem of Arithmetic.
Common mistakes when identifying prime and composite numbers
Even careful mathematicians can make mistakes. Here are some common errors and tips to avoid them.
- Correction: 1 has only one factor, so it is neither prime nor composite.
- Correction: Odd numbers like 9, 15, 21, 25, 27, 33 are composite because they have other factors.
- Correction: 2 is the smallest prime and the only even prime. It has exactly two factors: 1 and 2.
- Correction: Test it with small primes. For example, 91 is odd but 7 × 13 = 91, so it's composite.
Always check factors systematically. A quick way: if a number ends in 5 (and is greater than 5), it is composite because it is divisible by 5.
Reasoning through examples step by step
Let's walk through several numbers together and decide if they are prime or composite.
- Not even, so not divisible by 2.
- Sum of digits: 3+7=10, not divisible by 3.
- Does not end in 0 or 5, so not divisible by 5.
- Check 7: 7×5=35, 7×6=42, so no.
- No other primes to check (next prime 11: 11×3=33, 11×4=44, so no).
- Conclusion: 37 is prime.
- Not divisible by 2 (odd).
- Sum digits: 5+1=6, which is divisible by 3. So 51 ÷ 3 = 17.
- Therefore, 51 has factors 3 and 17 besides 1 and 51.
- Conclusion: 51 is composite.
- Not divisible by 2 (odd).
- Sum digits: 8+3=11, not divisible by 3.
- Ends with 3, not 0 or 5, so not divisible by 5.
- Check 7: 7×11=77, 7×12=84, so no.
- Check 11: 11×7=77, 11×8=88, so no.
- Conclusion: 83 is prime.
With practice, you will memorize many primes and recognize composites faster. Keep practicing with numbers up to 100.
Key ideas to remember
Here is a summary of everything you learned about prime and composite numbers.
- Prime numbers have exactly two factors: 1 and itself.
- Composite numbers have more than two factors.
- 0 and 1 are neither prime nor composite.
- 2 is the only even prime number.
- To test if a number is prime, check divisibility by smaller primes (2,3,5,7,11...).
- If you find any factor other than 1 and itself, it's composite.
Knowing prime and composite numbers helps you understand the structure of all whole numbers and prepares you for fractions, factors, and multiples.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.B.4 – Gain familiarity with factors and multiples. Identify prime and composite numbers.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.B.4. All content is 100% free; use it for whole-class instruction, independent study and practice, or homework.