Primes & Prime Factorization • Topic 2 of 4

Prime Factorization

To prime-factorize a number, divide repeatedly by the smallest prime that fits, then move up. For 360: 360 = 2·180 = 2·2·90 = 2·2·2·45 = 2·2·2·3·15 = 2·2·2·3·3·5, so 360 = 2³·3²·5. Writing the result in exponent form is the whole point — it is the "DNA" of the number, and every later question (counting factors, testing squares, finding GCD/LCM) reads off these exponents. A factor tree gives the same answer regardless of the order you split, because the factorization is unique. Keep the primes in increasing order so the exponents are easy to scan. On the GRE the numbers are chosen to factor cleanly, so this is fast by hand — reaching for the basic calculator here usually wastes time.

✅ Solved examples

1. Find the prime factorization of 84.
84 = 2·42 = 2·2·21 = 2·2·3·7. So 84 = 2²·3·7.
2. Find the prime factorization of 200.
200 = 2·100 = 2·2·50 = 2·2·2·25 = 2·2·2·5·5. So 200 = 2³·5².
3. Express 540 as a product of primes.
540 = 2·270 = 2·2·135 = 2·2·3·45 = 2·2·3·3·15 = 2·2·3·3·3·5. So 540 = 2²·3³·5.
4. Quantitative Comparison. Quantity A: the largest prime factor of 210. Quantity B: 5. Which is greater?
210 = 2·3·5·7, so its largest prime factor is 7. Quantity A (7) > Quantity B (5): Quantity A is greater.

✏️ Practice — try these, take hints as needed

1. Prime factorize 72.
Divide by 2 repeatedly.
72 = 8 × 9.
2³ and 3².
2³·3²
2. Prime factorize 126.
126 = 2 × 63.
63 = 9 × 7.
Collect primes.
2·3²·7
3. What is the largest prime factor of 156?
156 = 4 × 39.
39 = 3 × 13.
Largest prime.
13
4. Prime factorize 1000.
1000 = 10³.
10 = 2 × 5.
Cube it.
2³·5³
5. How many distinct prime factors does 330 have?
330 = 2 × 165.
165 = 3 × 5 × 11.
Count distinct.
4 (2, 3, 5, 11)

📝 Topic test — 8 questions

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