Prime factorization means rewriting a number as a product of prime numbers. It shows up in simplifying fractions, finding the LCM/GCF, working with radicals, and understanding divisibility. The good news: the process is always the same—break the number into factors until everything is prime, then rewrite repeated primes using exponents.
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A prime number has exactly two positive divisors: 1 and itself (2, 3, 5, 7, 11, 13, ...). A prime factorizationof a whole number is a multiplication expression where every factor is prime.
Example: 60 = 2·2·3·5 = 2^2·3·5.
Not a prime factorization: 60 = 6·10 (because 6 and 10 are not prime).
One important fact: for numbers greater than 1, the prime factorization is unique (aside from reordering). That’s why prime factorization is so useful for comparing numbers.
A factor tree is a way to keep splitting a number into two factors until every branch ends in primes. It doesn’t matter which factors you choose first—as long as you keep factoring composite numbers, you’ll end up with the same prime list.
Example: Prime factorize 84.
Notice the stop rule: you stop only when every factor is prime. If you still have 4, 6, 8, 9, 10, 12, etc., you are not done.
The division method means you repeatedly divide by prime numbers, starting with 2, then 3, then 5, and so on. This is especially nice for bigger numbers because it stays neat and reduces the number quickly.
Example: Prime factorize 360.
This method also makes it easy to count exponents: you used 2 three times, 3 twice, and 5 once.
You are finished when the remaining number is 1 (in the division method) or when every factor in your multiplication is prime (in the factor tree method). If you ever see a composite number still sitting in your final product, you need one more split.
Prime factorization is easier when you can quickly spot a small prime factor. You don’t need every divisibility rule—just a few reliable ones for the smallest primes.
Practical strategy: always try dividing by 2 first, then 3, then 5. If none work, try 7, 11, 13 next (especially for medium-sized numbers).
Example: Prime factorize 462.
The digit sum is 4 + 6 + 2 = 12, so 462 is divisible by 3.
Multiply-back check: 2·3·7·11 = 6·77 = 462. This kind of “mix of primes” is common in GCF/LCM problems.
Prime factors are like a “DNA code” for a number. Once two numbers are written in prime form, you can compare them by matching primes and exponents.
Example: Find the GCF of 84 and 360.
Take only primes that appear in both, using the smaller exponent. The same idea works for LCM, but you use the larger exponent.
Example: Find the LCM of 84 and 360.
For the LCM, you include every prime that appears in either number, using the larger exponent. This is the same logic as “take enough prime factors so that both original numbers divide it.”
If you get stuck, fall back on this routine. It keeps you from guessing random factors and helps you finish cleanly.
Writing 60 = 6·10 is not prime factorization. Keep going until every factor is prime.
72 = 2·2·2·3·3 is correct, but 72 = 2^3·3^2 is cleaner and easier to compare.
A prime factorization never includes 1. Using 1 just hides the real factors.
A quick multiply-back check catches missed primes or wrong exponents.
If you want a clear explanation of primes and factorization with worked examples, the short section in OpenStax Prime Factorization is a good reference.
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