Prime Numbers: Definition, List and How to Find Them
Part of the Math Concepts guide for Grades 3 to 8.
What Is a Prime Number?
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. It cannot be divided evenly by any other whole number. The first five prime numbers are 2, 3, 5, 7, and 11. A composite number is a whole number with more than two factors. The number 1 is neither prime nor composite; it sits in its own special category.
Prime vs. Composite Numbers
NumberPrime or Composite?Why1NeitherHas only one factor: itself2PrimeFactors: 1 and 2 only. The only even prime number.3PrimeFactors: 1 and 3 only4CompositeFactors: 1, 2, 45PrimeFactors: 1 and 5 only6CompositeFactors: 1, 2, 3, 6
Prime Numbers from 1 to 100
There are 25 prime numbers between 1 and 100:
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
A few things worth pointing out to students:
- 2 is the only even prime. Every other even number has 2 as a factor, so none of them can be prime.
- All prime numbers except 2 are odd.
- No prime number ends in 5 (except 5 itself), because any number ending in 5 is divisible by 5.
- Primes become less frequent as numbers get larger.
How to Tell If a Number Is Prime
The simplest method is called trial division. Here is how to use it step by step.
StepWhat to checkResult1Is the number less than 2?If yes, it is not prime2Is it 2 or 3?If yes, it is prime3Is it even (divisible by 2)?If yes, it is not prime4Try dividing by odd numbers: 3, 5, 7...Only test up to the square root of the number5No divisors found?The number is prime
Example: Is 37 prime?
- Find the square root of 37. It is approximately 6.1, so we only need to test primes up to 6: that means 2, 3, and 5.
- 37 divided by 2 = 18.5. Not a whole number, so 2 is not a factor.
- 37 divided by 3 = 12.3. Not a whole number, so 3 is not a factor.
- 37 divided by 5 = 7.4. Not a whole number, so 5 is not a factor.
- No divisors found, so 37 is prime.
Common Mistakes to Avoid
Mistake 1: Thinking 1 is a prime number
This is the most common mistake I see. By definition, a prime number must have exactly two distinct factors. The number 1 has only one factor, which is itself. Since it does not meet the two-factor requirement, 1 is neither prime nor composite.
Mistake 2: Forgetting that 2 is prime
Some students assume all prime numbers are odd, which leads them to skip 2. Remember: 2 is prime because its only factors are 1 and 2. It is the only even prime number.
Mistake 3: Confusing prime and composite
A helpful reminder: the word composite means made up of parts. A composite number can be built by multiplying smaller whole numbers together. A prime number cannot be broken down that way.
Common Questions About Prime Numbers
What is a prime number?
A prime number is a whole number greater than 1 with exactly two factors: 1 and itself. It cannot be divided evenly by any other number. Examples include 2, 3, 5, 7, 11, and 13.
Is 1 a prime number?
No. The number 1 is not prime. A prime number must have exactly two distinct factors. Since 1 has only one factor, it does not meet that requirement. It is neither prime nor composite.
What is the smallest prime number?
2 is the smallest prime number. It is also the only even prime number. Every other even number has 2 as a factor, giving it at least three factors and making it composite.
What is the difference between prime and composite numbers?
A prime number has exactly two factors: 1 and itself. A composite number has more than two factors and can be divided evenly by numbers other than 1 and itself. The number 1 is neither prime nor composite.
How many prime numbers are there between 1 and 100?
There are 25 prime numbers between 1 and 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.
Why are prime numbers important?
Prime numbers are fundamental in mathematics and cryptography. Every composite number can be expressed as a unique product of prime numbers, which is called prime factorization. Modern internet encryption, including the HTTPS protocol that keeps websites secure, relies on the difficulty of factoring very large numbers into their prime components.
