Every divisibility relationship involves two numbers playing opposite roles. When 3∣12, the number 3 is a factor — it divides in. The number 12 is a multiple — it is divided into. Factors look downward, toward the smaller numbers that build a given integer. Multiples look upward, toward the infinite sequence of numbers that a given integer builds. Together they form the structural vocabulary of divisibility.
Factors (Divisors)
A factor of n is any integer a such that a∣n — meaning n=a⋅k for some integer k. The terms "factor" and "divisor" are interchangeable.
Every positive integer has at least two factors: 1 and itself. The number 1 divides everything, and every number divides itself. Most numbers have additional factors between these two extremes.
The factors of 24 are 1,2,3,4,6,8,12,24 — eight in total. The factors of 13 are just 1 and 13 — making it prime.
Unlike multiples, which extend infinitely, the factor set of any positive integer is finite. No factor of n can exceed n itself (in absolute value), so the search space is bounded from the start.
Finding All Factors
A systematic search tests each integer from 1 up to n. If a divides n, both a and an are factors — a pair produced from a single test. Stopping at n is sufficient because if a>n, then an<n, and that smaller partner would already have been found.
For n=36, test integers from 1 to 6 (since 36=6):
Every factor a of n has a partner: an. Together they satisfy a⋅an=n, and listing factor pairs ensures no divisor is overlooked.
For n=24, the pairs are: (1,24), (2,12), (3,8), (4,6). Each pair multiplies to 24, and together they account for all eight factors.
For perfect squares, one pair collapses into a repeated value. The number 36 has the pair (6,6) — the square root paired with itself. This is why perfect squares have an odd number of factors: every other factor appears in a pair of two distinct values, but the square root stands alone.
Organizing factors into pairs is more reliable than listing them individually. Starting from (1,n) and working inward, each successful division test produces two factors at once — halving the work and providing a built-in completeness check.
Proper Divisors
The proper divisors of n are all of its factors except n itself. They are the numbers that divide n without being equal to it.
The proper divisors of 12 are {1,2,3,4,6}. The number 12 itself is excluded.
Proper divisors define three classical categories. A number is perfect if the sum of its proper divisors equals the number: 6=1+2+3, so 6 is perfect. A number is abundant if the sum exceeds it: the proper divisors of 12 sum to 1+2+3+4+6=16>12. A number is deficient if the sum falls short: the proper divisors of 10 sum to 1+2+5=8<10.
Perfect numbers are rare. The first four are 6,28,496, and 8,128. Every known perfect number is even, and whether an odd perfect number exists remains one of the oldest unsolved problems in mathematics.
Multiples
A multiple of a is any number b such that a∣b — meaning b=a⋅k for some positive integer k. Where factors divide into a number, multiples are produced by multiplying out from it.
The multiples of 7 are 7,14,21,28,35,42,… — an infinite sequence with no largest member. Every positive integer generates infinitely many multiples.
Zero is a multiple of every number, since 0=a⋅0 for any a. It sits at the start of every multiple sequence, though it is often omitted when listing "the multiples of a" in contexts where only positive values are relevant.
The modulo operation identifies which multiple of n is closest to a given number a without exceeding it: a=n⋅q+r, where n⋅q is that nearest multiple and r is the gap between it and a.
Factors vs Multiples
The two concepts are inverses of each other, and mixing them up is one of the most common errors in elementary number theory.
Factors go downward. The factors of 24 are 1,2,3,4,6,8,12,24 — all less than or equal to 24. There are finitely many of them.
Multiples go upward. The multiples of 24 are 24,48,72,96,… — all greater than or equal to 24 (excluding zero). There are infinitely many.
The relationship is symmetric: a is a factor of b if and only if b is a multiple of a. The number 3 is a factor of 12, and 12 is a multiple of 3. Same fact, opposite viewpoints.
A useful mnemonic: factors fit into a number (they are smaller), multiples multiply out from a number (they are larger). The factor set is contained; the multiple set is unbounded.
Common Factors
A common factor of two numbers m and n is a number that divides both. Finding common factors means intersecting the two factor sets.
The factors of 12 are {1,2,3,4,6,12}. The factors of 18 are {1,2,3,6,9,18}. The common factors — numbers appearing in both sets — are {1,2,3,6}.
Every pair of positive integers shares at least one common factor: 1 divides both, always. The question is whether they share anything larger.
The greatest common factor is the largest member of the common factor set. For 12 and 18, it is 6. This value — the GCD — plays a central role in simplifying fractions, solving equations, and determining whether two numbers are coprime.
A common multiple of two numbers m and n is a number divisible by both. Where common factors are drawn from finite sets, common multiples form an infinite sequence.
The multiples of 4 are 4,8,12,16,20,24,28,32,36,… The multiples of 6 are 6,12,18,24,30,36,… The common multiples are 12,24,36,48,… — every value that appears in both lists.
The smallest positive common multiple is the LCM. For 4 and 6, it is 12. Every other common multiple is a multiple of the LCM: 24=2⋅12, 36=3⋅12, and so on.
Common multiples arise naturally in problems involving synchronization. If one event repeats every 4 days and another every 6 days, they coincide every 12 days — the LCM of their periods.
For small numbers, counting factors by listing them is practical. For larger numbers, prime factorization provides a formula.
If n=p1a1⋅p2a2⋯pkak, the number of positive divisors of n is:
(a1+1)(a2+1)⋯(ak+1)
Each prime factor pi can appear in a divisor 0,1,2,…,ai times — that is ai+1 independent choices. The total number of divisors is the product of all these choices.
The number 72=23⋅32 has (3+1)(2+1)=12 factors. Listing them confirms: 1,2,3,4,6,8,9,12,18,24,36,72 — exactly twelve.
The number 100=22⋅52 has (2+1)(2+1)=9 factors.
The formula reveals structure that raw listing obscures. A number with many small prime factors has more divisors than a number of similar size with fewer, larger prime factors. The number 60=22⋅3⋅5 has 12 factors, while 64=26 has only 7.
The sum of all positive divisors of n is denoted σ(n). Like the divisor count, it can be computed directly from the prime factorization.
For n=12: the divisors are 1,2,3,4,6,12, and σ(12)=1+2+3+4+6+12=28.
For n=6: the divisors are 1,2,3,6, and σ(6)=1+2+3+6=12. The sum of all divisors is twice the number — but more revealing is that the sum of the proper divisors is 1+2+3=6, equal to n itself. This makes 6 a perfect number.
The relationship between σ(n) and n classifies every positive integer. When σ(n)−n=n, the number is perfect. When σ(n)−n>n, it is abundant. When σ(n)−n<n, it is deficient. Most numbers are deficient; abundance and perfection are the exceptions.
Worked Examples
Find all factors of 48. Test from 1 to 48≈6.9: 1∣48 (pair 1,48), 2∣48 (pair 2,24), 3∣48 (pair 3,16), 4∣48 (pair 4,12), 5∤48, 6∣48 (pair 6,8). Factors: {1,2,3,4,6,8,12,16,24,48} — ten in total.
List the first 8 multiples of 9: 9,18,27,36,45,54,63,72.
Find the common factors of 30 and 45. Factors of 30: {1,2,3,5,6,10,15,30}. Factors of 45: {1,3,5,9,15,45}. Common: {1,3,5,15}. The GCD is 15.
Count the factors of 180=22⋅32⋅5. Formula: (2+1)(2+1)(1+1)=18 factors.
Is 28 perfect? Proper divisors: 1,2,4,7,14. Sum: 1+2+4+7+14=28. Yes — 28 is the second perfect number.
Is 20 abundant or deficient? Proper divisors: 1,2,4,5,10. Sum: 22>20. Abundant.