Factors and Multiples • Topic 3 of 7

Prime Factorization

Prime factorization expresses a number as a product of primes, for example 60 = 2² × 3 × 5. By the Fundamental Theorem of Arithmetic this expression is unique apart from order. Find it by repeatedly dividing out the smallest prime, or with a factor tree. Prime factorization powers many SAT shortcuts: the number of divisors equals the product of (each exponent + 1); GCF and LCM come straight from the shared and combined prime powers; and a perfect square has only even exponents. Writing a number in this form turns hard divisibility questions into simple bookkeeping with exponents.

✅ Solved examples

1. Find the prime factorization of 84.
Divide out smallest primes: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7, and 7 is prime. So 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
2. How many positive divisors does 72 have?
72 = 2³ × 3². Add one to each exponent and multiply: (3+1)(2+1) = 4 × 3 = 12 divisors.
3. Is 200 a perfect square? Use prime factorization.
200 = 2³ × 5². A perfect square needs every exponent even. Here the exponent of 2 is 3 (odd), so 200 is not a perfect square.
4. Express 360 as a product of primes.
360 ÷ 2 = 180, ÷2 = 90, ÷2 = 45, ÷3 = 15, ÷3 = 5, and 5 is prime. So 360 = 2³ × 3² × 5.

✏️ Practice — try these, take hints as needed

1. Find the prime factorization of 126.
Start by dividing by the smallest prime, 2.
126 = 2 × 63; now factor 63.
63 = 7 × 9 = 7 × 3².
126 = 2 × 3² × 7.
2. How many positive divisors does 2³ × 5² have?
Use the divisor-counting rule on the exponents.
Add 1 to each exponent: (3+1) and (2+1).
Multiply the results.
12 divisors.
3. What is the smallest number you can multiply 90 by to get a perfect square?
Factor 90 = 2 × 3² × 5.
A perfect square needs all even exponents; find the odd ones.
2 and 5 have exponent 1 — multiply by 2 × 5.
10 (since 90 × 10 = 900 = 30²).
4. Find the prime factorization of 1000.
1000 = 10³.
Write 10 = 2 × 5.
So 10³ = (2 × 5)³.
1000 = 2³ × 5³.
5. A number is 2⁴ × 3. How many of its divisors are even?
Total divisors = (4+1)(1+1) = 10.
Odd divisors use only the factor 3: that is 3⁰ and 3¹ → 2 of them.
Even divisors = total − odd.
8 even divisors.

📝 Topic test — 8 questions

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