Primes & Prime Factorization • Topic 3 of 4

Counting Factors

Never list divisors on the GRE — count them from the prime factorization. If n = p^a · q^b · r^c, then the number of positive factors is (a + 1)(b + 1)(c + 1). The logic: each factor chooses each prime's exponent independently from 0 up to its max, giving (a+1) choices for p, and so on. For 72 = 2³·3², the count is (3+1)(2+1) = 12 factors. A beautiful corollary the GRE tests directly: a number has an odd number of factors if and only if it is a perfect square, because only then does every (exponent + 1) term multiply out to an odd total. This single formula turns "how many divisors?" — a question that looks like tedious listing — into one multiplication you can do in your head.

✅ Solved examples

1. How many positive factors does 36 have?
36 = 2²·3². Count = (2+1)(2+1) = 9. (They are 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine, confirming an odd count for the perfect square 36.)
2. How many positive factors does 240 have?
240 = 2⁴·3·5. Count = (4+1)(1+1)(1+1) = 5 × 2 × 2 = 20 factors.
3. How many odd factors does 180 have?
180 = 2²·3²·5. Odd factors use no 2, so only 3²·5 contributes: (2+1)(1+1) = 6 odd factors.
4. Quantitative Comparison. Quantity A: the number of positive factors of 100. Quantity B: the number of positive factors of 99. Which is greater?
100 = 2²·5² → (2+1)(2+1) = 9. 99 = 3²·11 → (2+1)(1+1) = 6. Quantity A (9) is greater.

✏️ Practice — try these, take hints as needed

1. How many positive factors does 48 have?
48 = 2⁴·3.
(4+1)(1+1).
Multiply.
10
2. How many positive factors does 2·3·5·7 have?
Each exponent is 1.
(1+1)⁴.
2⁴.
16
3. How many factors does 64 have?
64 = 2⁶.
(6+1).
Single prime.
7
4. How many odd factors does 360 have?
360 = 2³·3²·5.
Ignore the 2s.
(2+1)(1+1).
6
5. A number has exactly 3 factors. What kind of number is it?
(a+1) = 3 needs a = 2.
Single prime squared.
Like 4, 9, 25.
Square of a prime

📝 Topic test — 8 questions

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