Prime Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Prime Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Integers greater than 1 whose only positive divisors are 1 and themselves—they cannot be factored further.

Primes can't be broken down further—they're the 'atoms' of multiplication.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Every number is either prime or can be factored into primes uniquely.

Common stuck point: 1 is NOT prime—primes need exactly two distinct factors. And 2 is the only even prime; every other even number has 2 as a factor.

Sense of Study hint: Try dividing the number by every prime up to its square root (2, 3, 5, 7...). If none divide evenly, the number is prime.

Worked Examples

Example 1

easy
Find the prime factorization of 180.

Solution

  1. 1
    Divide by 2: 180 \div 2 = 90. Divide again: 90 \div 2 = 45.
  2. 2
    45 is odd, try 3: 45 \div 3 = 15. Again: 15 \div 3 = 5.
  3. 3
    5 is prime, so stop. Prime factorization: 180 = 2^2 \times 3^2 \times 5.

Answer

2^2 \times 3^2 \times 5
To find the prime factorization, repeatedly divide by the smallest prime that divides evenly, working upward. Every composite number has a unique prime factorization (Fundamental Theorem of Arithmetic).

Example 2

medium
Determine whether 97 is prime.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
List all prime numbers between 20 and 40.

Example 2

easy
Is 51 prime or composite?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

factorsdivisibility intuition