Prime Numbers and Prime Factorization

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. A composite number has more than two factors. Prime factorization breaks any composite number down into a product of primes — like finding the atomic building blocks of a number.

Prime vs. Composite

Special cases:

• 1 is neither prime nor composite (it has only one factor)
• 2 is the only even prime number (every other even number is divisible by 2)

Prime Numbers Under 100

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$

That is 25 prime numbers under 100. The rest (other than 1) are composite.

How to Test if a Number Is Prime

To check whether a number $n$ is prime, try dividing by every prime up to $\sqrt{n}$. If none divide evenly, the number is prime.

Example 1: Is 29 prime?

$\sqrt{29} \approx 5.4$, so check primes up to 5: 2, 3, 5.

• $29 \div 2 = 14.5$ — not divisible
• $29 \div 3 \approx 9.67$ — not divisible
• $29 \div 5 = 5.8$ — not divisible

No prime factor found. 29 is prime.

Example 2: Is 51 prime?

$\sqrt{51} \approx 7.1$, so check primes up to 7: 2, 3, 5, 7.

• $51 \div 2 = 25.5$ — no
• $51 \div 3 = 17$ — yes!

$51 = 3 \times 17$. 51 is composite.

Prime Factorization

Every composite number can be written as a product of prime numbers. This representation is unique (up to the order of the factors) — this is called the Fundamental Theorem of Arithmetic.

The Factor Tree Method

1. Write the number at the top
2. Split it into any two factors
3. If a factor is prime, circle it (it is done)
4. If a factor is composite, split it again
5. Continue until all branches end in primes

Example 3: Prime Factorization of 60

Start: $60 = 6 \times 10$

Split 6: $6 = 2 \times 3$ (both prime)

Split 10: $10 = 2 \times 5$ (both prime)

$60 = 2 \times 3 \times 2 \times 5 = 2^2 \times 3 \times 5$

Example 4: Prime Factorization of 72

Start: $72 = 8 \times 9$

Split 8: $8 = 2 \times 4 = 2 \times 2 \times 2$

Split 9: $9 = 3 \times 3$

$72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$

Example 5: Prime Factorization of 180

Start: $180 = 18 \times 10$

$18 = 2 \times 9 = 2 \times 3 \times 3$

$10 = 2 \times 5$

$180 = 2 \times 3 \times 3 \times 2 \times 5 = 2^2 \times 3^2 \times 5$

The Division Method (Alternative)

Divide by the smallest prime that works, then divide the quotient, and repeat:

$180 \div 2 = 90$ $90 \div 2 = 45$ $45 \div 3 = 15$ $15 \div 3 = 5$ $5 \div 5 = 1$

$180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^2 \times 3^2 \times 5$

Both methods give the same result.

Why Prime Factorization Matters

Prime factorization is the key to:

• Finding the GCF and LCM of two or more numbers
• Simplifying fractions efficiently
• Understanding number relationships in algebra

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Prime numbers have exactly 2 factors (1 and themselves); composite numbers have more
• 1 is neither prime nor composite; 2 is the only even prime
• To test for primality, check divisibility by primes up to $\sqrt{n}$
• Prime factorization expresses any composite number as a product of primes
• Use the factor tree or division method — both give the same result
• Every number has exactly one prime factorization (Fundamental Theorem of Arithmetic)

Return to Arithmetic for more foundational math topics.