Divisibility, Factors & Multiples • Topic 2 of 5

Factors & Multiples

A factor (or divisor) of n divides n with no remainder; a multiple of n is n times an integer. Factors of a positive integer are finite and paired — for 24 the pairs are (1,24), (2,12), (3,8), (4,6) — while multiples go on forever: 24, 48, 72, … . The GRE leans on three facts. First, the smallest factor of any n > 1 is 1 and the largest is n itself. Second, every factor of n is at most n, and every multiple of n is at least n (for positive n). Third, if a is a factor of b and b is a factor of c, then a is a factor of c. A common wording trap: "factor" and "multiple" are opposites — 4 is a factor of 12, and 12 is a multiple of 4. Read which direction the question wants before you answer.

✅ Solved examples

1. List all positive factors of 36.
Pair them up: 1×36, 2×18, 3×12, 4×9, 6×6. So the factors are 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors (the repeated 6 counts once, because 36 is a perfect square).
2. What is the smallest multiple of 7 that is greater than 100?
100 ÷ 7 ≈ 14.3, so 7 × 15 = 105 is the first multiple above 100.
3. If 12 is a factor of n, list three facts that must be true about n.
n is a multiple of 12, so n is also divisible by every factor of 12 — in particular by 2, 3, 4 and 6. And n ≥ 12 (for positive n).
4. Quantitative Comparison. Quantity A: the number of positive factors of 12. Quantity B: the number of positive factors of 18. Which is greater?
Factors of 12: 1,2,3,4,6,12 → 6 factors. Factors of 18: 1,2,3,6,9,18 → 6 factors. They are equal.

✏️ Practice — try these, take hints as needed

1. List all positive factors of 28.
Pair them.
1×28, 2×14, 4×7.
Collect all.
1, 2, 4, 7, 14, 28
2. What is the largest factor of 45 other than 45 itself?
Largest proper factor = n / smallest prime.
Smallest prime factor is 3.
45 / 3.
15
3. Smallest multiple of 9 greater than 50?
50 / 9 ≈ 5.6.
Next integer is 6.
9 × 6.
54
4. How many multiples of 6 are there between 1 and 50?
6, 12, ... , 48.
48 / 6 = 8.
Count them.
8
5. If n is a multiple of 15, must it be a multiple of 5?
5 is a factor of 15.
Factor chain rule.
Yes it must.
Yes

📝 Topic test — 8 questions

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