Factors & Multiples
1. Core Idea
If a x b = N with a, b integers, then a and b are factors of N, and N is a multiple of a and b.
Factors are at most N and finite in number. Multiples are at least N and infinite. Students who keep those two sentences straight avoid half the errors in this topic.
2. Must-Know Rules
| Concept | Rule |
|---|---|
| Factor | divides N exactly, leaving remainder 0 |
| Multiple | N x k for integer k |
| Smallest factor of N > 1 | its smallest prime factor |
| Largest factor of N | N itself |
| Smallest positive multiple of N | N itself |
| Number of factors | if N = p^a x q^b x r^c, count = (a+1)(b+1)(c+1) |
| Sum of factors | (p^0+...+p^a)(q^0+...+q^b)... |
| Factors come in pairs | d and N/d |
| Perfect squares | have an ODD number of factors |
Why perfect squares are odd-count. Factors pair up as d and N/d. For a perfect square, the middle pair is the square root paired with itself, so one factor is unpaired. 36 has factors 1,2,3,4,6,9,12,18,36 — nine of them.
Counting factors — worked mechanic.
72 = 2^3 x 3^2
Number of factors = (3+1)(2+1) = 4 x 3 = 12.
Check: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Twelve. Correct.
Listing factors efficiently. Walk upward from 1 to sqrt(N), and each divisor you find gives you its partner for free.
3. Worked Examples
Example 1 — Number of factors
How many positive factors does 180 have?
180 = 2^2 x 3^2 x 5^1
Count = (2+1)(2+1)(1+1) = 3 x 3 x 2 = 18.
Example 2 — Even factors only
How many even factors does 180 have?
An even factor must contain at least one 2. Fix one 2 and count the factors of what remains:
180/2 = 90 = 2^1 x 3^2 x 5
Number of factors of 90 = (1+1)(2+1)(1+1) = 12.
12 even factors. (And 18 - 12 = 6 odd factors — check: the odd factors are the factors of 3^2 x 5, which is (2+1)(1+1) = 6. Correct.)
Example 3 — Multiples in a range
How many multiples of 7 are there between 100 and 300 inclusive?
Smallest multiple >= 100: 7 x 15 = 105.
Largest multiple <= 300: 7 x 42 = 294.
Count = 42 - 15 + 1 = 28.
4. GRE Traps
- Forgetting 1 and N. Both are factors of N.
- Counting only prime factors when the question asks for all factors.
- Negative factors. Unless the question says "positive", technically -3 is a factor of 12. The GRE almost always means positive — but read carefully.
- Off-by-one in counting multiples. The count in a range is (last - first)/step + 1. The "+1" is forgotten constantly.
- Assuming a factor of a product must be a factor of one of the parts. 6 is a factor of 4 x 9 = 36 but not of 4 or 9.
- Perfect-square factor count. If a question says "N has an odd number of factors", it is telling you N is a perfect square.
5. Speed Tricks
- Prime factorise first, always. Every factor question reduces to exponents once you do.
- (a+1)(b+1)(c+1) for factor counts — memorise the "+1 to each exponent" rule.
- Odd factors = drop all the 2s, then count. Even factors = total minus odd.
- Factor pairs halve your listing work — stop at sqrt(N).
- "How many multiples of k in [a, b]" = floor(b/k) - floor((a-1)/k).
6. Self-Check
Q1. How many positive factors does 96 have?
Q2. How many multiples of 12 are between 50 and 200 inclusive?
Q3. If a number has exactly 3 positive factors, what kind of number is it?
Answers
A1. 96 = 2^5 x 3, so (5+1)(1+1) = 12.
A2. First: 60 (12x5). Last: 192 (12x16). Count = 16 - 5 + 1 = 12.
A3. (a+1) = 3 means a = 2 and only one prime, so it is the square of a prime (4, 9, 25, 49, ...).
7. One-Line Summary for the Board
Prime factorise, then add 1 to every exponent and multiply. Factors are finite; multiples are infinite.