A prime number is a whole number greater than 1 with exactly two positive divisors: 1 and itself. The numbers 2, 3, 5, 7 and 11 are prime. The number 9 is not, because 3 divides it exactly. To check a number, look for a divisor other than 1 and the number itself; finding just one proves it is composite.
You do not need to memorize a long list or divide by every smaller number. A few quick checks, followed by a clear stopping rule, will handle the numbers you are likely to meet in homework and everyday puzzles.
What does “divides exactly” mean?
A divisor, also called a factor, goes into a number with no remainder. For example, 4 is a divisor of 28 because 28 ÷ 4 = 7. But 5 is not: five groups of five use 25, leaving 3 over. For prime-number questions, the quotient must be a whole number, not a decimal.
Compare 17 with 18. The positive divisors of 17 are just 1 and 17, so it is prime. The divisors of 18 are 1, 2, 3, 6, 9 and 18, so it is composite. “Composite” means a whole number greater than 1 that has additional factors. This matches the definition in EDC’s guide to prime numbers.
Why is 1 not prime, and why is 2 prime?
The number 1 has only one positive divisor: 1. Writing 1 × 1 does not produce two different divisors. It therefore fails the requirement for exactly two and is neither prime nor composite.
The number 2 has exactly two positive divisors, 1 and 2, so it is prime. Every larger even number also has 2 as a divisor in addition to 1 and itself. That makes 2 the only even prime.
Under the usual elementary definition, zero and negative integers are not prime either. Primes are positive whole numbers greater than 1. Fractions and non-whole decimals are outside this definition. Wolfram MathWorld’s prime-number entry explains this scope and the special case of 1.
How to test whether a number is prime
First, check that the number is greater than 1. Then try the small prime divisors in order: 2, 3, 5, 7, 11 and so on. Stop immediately if one divides exactly. Otherwise, continue only through the square root of the number, including the square root when it is an integer.
These quick checks save arithmetic: an even number greater than 2 is composite; a number greater than 3 whose digits add to a multiple of 3 is composite; and a number greater than 5 ending in 0 or 5 is composite. Passing these checks does not automatically make a number prime. For instance, 77 passes all three but equals 7 × 11.
You can test every integer from 2 through the square root if that feels simpler. Testing only primes is faster because a composite divisor would contain a smaller prime factor that you already checked.
Worked examples: 29, 49 and 91
Is 29 prime? Its square root lies between 5 and 6, so check 2, 3 and 5. It is odd, its digit sum is 11, and it does not end in 0 or 5. None divides exactly, so 29 is prime. There is no need to try 7 or every number up to 28.
Is 49 prime? It is odd, its digit sum is 13, and it does not end in 0 or 5. But the square root of 49 is exactly 7. You must include that final check: 49 ÷ 7 = 7. Therefore, 49 is composite. Its distinct positive divisors are 1, 7 and 49.
Is 91 prime? Its square root lies between 9 and 10, so the necessary prime divisors are 2, 3, 5 and 7. The first three fail, but 91 ÷ 7 = 13. Finding that factor is enough to conclude that 91 is composite.
Why can you stop at the square root?
Factors come in pairs. If a number equals a × b and both factors were larger than its square root, their product would be larger than the original number. Therefore, a composite number must have a factor no larger than its square root. Checking that smaller side catches the pair.
For example, 143 = 11 × 13. Its square root is between 11 and 12, so checking through 11 finds the factor even though 13 is beyond the stopping point. This rule makes trial division manageable for small numbers; very large numbers need more efficient methods.
How to find every prime up to 100
Use the sieve of Eratosthenes when you want a whole list rather than a single answer. Write the integers from 2 to 100 on paper. Keep 2 and cross out its larger multiples. Keep 3 and cross out its larger multiples. Repeat with 5 and 7, always keeping the prime itself. Ignore numbers already crossed out.
You can stop there because the square root of 100 is 10. Every composite number up to 100 has a prime factor among 2, 3, 5 and 7. For printable grids and a related investigation, see NRICH’s sieve activity.
The 25 numbers left 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 and 97.
Quick questions and a practice check
Are all odd numbers prime? No. For example, 15 = 3 × 5 and 21 = 3 × 7. Oddness removes the factor 2, but it does not remove other possibilities.
Can a square number be prime? No square of a whole number greater than 1 is prime, because that whole number is a factor. For example, 121 = 11 × 11. The remaining nonnegative squares, 0 and 1, are not prime either.
Try 31, 57 and 121. The answers are prime, composite and composite. For 31, check 2, 3 and 5. For 57, the digits sum to 12, so 3 divides it: 57 = 3 × 19. For 121, include 11, its square root.
For another arithmetic skill with worked examples, continue with how to calculate percentages.
Optional supplies for hands-on practice
Paper and pencil are enough. If you prefer physical objects, you can browse counting tiles on Amazon to experiment with equal groups.
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