Understanding Factors and Prime Numbers
The positive factors of an integer \( n > 0 \) are all positive integers that divide \( n \) with no remainder. For example, the positive factors of \( 12 \) are \( 1, 2, 3, 4, 6, \) and \( 12 \), because:
\[ 12 = 2 \times 6 = 3 \times 4 = 1 \times 12 \]Among these, the numbers \( 2 \) and \( 3 \) are prime; \( 4, 6, \) and \( 12 \) are not. That distinction is what prime factorization is about.
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every integer greater than \( 1 \) can be written as a product of prime numbers in exactly one way, up to the order of the factors.
A prime number is an integer greater than \( 1 \) whose only positive divisors are \( 1 \) and itself. The first few primes are:
\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, \ldots \]Note: \( 1 \) is not prime (it has only one positive divisor), and neither is \( 0 \).
How to find prime factors: Divide your number by \( 2 \) as many times as possible, then by \( 3 \), then \( 5 \), then \( 7 \), and so on — always using primes and stopping when the remaining quotient is itself prime.
Worked Examples
Example 2 — Factor 20
\[ \begin{aligned} 20 &= 2 \times 10 \\ &= 2 \times 2 \times 5 \\ &= 2^2 \times 5 \end{aligned} \]Example 3 — Factor 100
\[ \begin{aligned} 100 &= 2 \times 50 \\ &= 2 \times 2 \times 25 \\ &= 2^2 \times 5^2 \end{aligned} \]Example 4 — Factor 1020
\[ \begin{aligned} 1020 &= 2 \times 510 \\ &= 2 \times 2 \times 255 \\ &= 2^2 \times 3 \times 85 \\ &= 2^2 \times 3 \times 5 \times 17 \end{aligned} \]Example 5 — Factor 634
\[ 634 = 2 \times 317 \]Note: \( 317 \) is prime (it is not divisible by any prime up to \( \sqrt{317} \approx 17.8 \)), so the factorization stops here.
Example 6 — Factor 720
\[ \begin{aligned} 720 &= 2 \times 360 \\ &= 2^2 \times 180 \\ &= 2^3 \times 90 \\ &= 2^4 \times 45 \\ &= 2^4 \times 3^2 \times 5 \end{aligned} \]Interactive Practice Questions
Part A: Identifying Valid Prime Factorizations
Decide whether each expression is a correct prime factorization, then expand View Solution to check your reasoning.
a) \( 2 \times 2 \times 4 \)
Not a prime factorization. The factor \( 4 \) is not prime (\( 4 = 2^2 \)).
b) \( 3 \times 3 \times 5 \times 9 \)
Not a prime factorization. The factor \( 9 \) is not prime (\( 9 = 3^2 \)).
c) \( 3 \times 3 \times 5 \times 17 \)
Valid prime factorization. All factors \( 3, 5, \) and \( 17 \) are prime. This equals \( 3^2 \times 5 \times 17 = 765 \).
d) \( 2 \times 5 \times 5 \times 21 \)
Not a prime factorization. The factor \( 21 \) is not prime (\( 21 = 3 \times 7 \)). The correct factorization would be \( 2 \times 3 \times 5^2 \times 7 = 1050 \).
e) \( 2 \times 2 \times 3 \times 5 \times 41 \)
Valid prime factorization. All factors \( 2, 3, 5, \) and \( 41 \) are prime. This equals \( 2^2 \times 3 \times 5 \times 41 = 2460 \).
Part B: Factor Each Number into Primes
Write each number as a product of primes.
Factor \( 18 \)
\[ 18 = 2 \times 9 = 2 \times 3^2 \]
Factor \( 300 \)
\[ 300 = 3 \times 100 = 3 \times 2^2 \times 5^2 = 2^2 \times 3 \times 5^2 \]
Factor \( 123 \)
\[ 123 = 3 \times 41 \]
Factor \( 1200 \)
\[ 1200 = 12 \times 100 = (3 \times 2^2) \times (2^2 \times 5^2) = 2^4 \times 3 \times 5^2 \]
Factor \( 1450 \)
\[ 1450 = 10 \times 145 = (2 \times 5) \times (5 \times 29) = 2 \times 5^2 \times 29 \]
Common Prime Factorizations
| Number | Prime Factorization |
|---|---|
| 12 | 2² × 3 |
| 18 | 2 × 3² |
| 20 | 2² × 5 |
| 36 | 2² × 3² |
| 60 | 2² × 3 × 5 |
| 100 | 2² × 5² |
| 300 | 2² × 3 × 5² |
| 720 | 2⁴ × 3² × 5 |
| 1200 | 2⁴ × 3 × 5² |
Frequently Asked Questions
What is prime factorization?
Prime factorization is the process of writing an integer greater than \( 1 \) as a product of prime numbers. By the Fundamental Theorem of Arithmetic, this factorization is unique up to the order of the factors.
What is the prime factorization of 12?
\( 12 = 2 \times 2 \times 3 = 2^2 \times 3 \).
What is the prime factorization of 100?
\( 100 = 2 \times 2 \times 5 \times 5 = 2^2 \times 5^2 \).
What is the prime factorization of 720?
\( 720 = 2^4 \times 3^2 \times 5 \).
Is 1 prime?
No. A prime has exactly two positive divisors: \( 1 \) and itself. The number \( 1 \) has only one positive divisor, so it is neither prime nor composite, and it has no prime factorization.
Is 0 prime?
No. Primes are strictly greater than \( 1 \), so \( 0 \) is neither prime nor composite and has no prime factorization.
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Last updated: October 17, 2026