What is a factor of a number?

A factor (or divisor) of n is any integer a such that a divides n evenly — meaning n = a × k for some integer k. Every positive integer has at least two factors: 1 and itself. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

What is a multiple of a number?

A multiple of a is any number b such that a divides b — meaning b = a × k for some positive integer k. The multiples of 7 are 7, 14, 21, 28, 35... — an infinite sequence. Zero is technically a multiple of every number.

What is the difference between factors and multiples?

Factors go downward (smaller than or equal to the number) and are finite. Multiples go upward (larger than or equal) and are infinite. If a is a factor of b, then b is a multiple of a. For 12: factors are 1,2,3,4,6,12; multiples are 12,24,36,48...

How do you find all factors of a number?

Test integers from 1 to √n. If a divides n, both a and n/a are factors — a pair from one test. For 36: test 1-6, finding pairs (1,36), (2,18), (3,12), (4,9), (6,6). This gives all nine factors efficiently.

What are factor pairs?

Every factor a of n has a partner n/a, and together they multiply to n. For 24: pairs are (1,24), (2,12), (3,8), (4,6). For perfect squares, the square root pairs with itself, giving an odd number of total factors.

What are proper divisors?

Proper divisors of n are all factors except n itself. For 12: proper divisors are {1, 2, 3, 4, 6}. The sum of proper divisors determines whether a number is perfect (sum = n), abundant (sum > n), or deficient (sum < n).

What is a perfect number?

A perfect number equals the sum of its proper divisors. The number 6 is perfect: 1 + 2 + 3 = 6. The number 28 is perfect: 1 + 2 + 4 + 7 + 14 = 28. Perfect numbers are rare — the first four are 6, 28, 496, and 8128.

What are common factors?

Common factors of two numbers m and n are numbers that divide both. For 12 and 18: factors of 12 are {1,2,3,4,6,12}, factors of 18 are {1,2,3,6,9,18}, common factors are {1,2,3,6}. The largest common factor is the GCD.

What are common multiples?

Common multiples of m and n are numbers divisible by both. For 4 and 6: common multiples are 12, 24, 36, 48... The smallest positive common multiple is the LCM. Every common multiple is a multiple of the LCM.

How do you count the number of factors?

Use prime factorization. If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, the number of factors is (a₁+1)(a₂+1)...(aₖ+1). For 72 = 2³ × 3²: (3+1)(2+1) = 12 factors. Each exponent plus one represents choices for that prime.

Why do perfect squares have an odd number of factors?

Factors come in pairs (a, n/a) that multiply to n. For perfect squares, the square root pairs with itself (√n, √n), counting as one factor instead of two. All other pairs contribute two factors, so the total is odd.

What is the sum of divisors function σ(n)?

σ(n) is the sum of all positive divisors of n. For 12: σ(12) = 1+2+3+4+6+12 = 28. This function determines perfect/abundant/deficient classification: perfect when σ(n) = 2n, abundant when σ(n) > 2n, deficient when σ(n) < 2n.

Factors and Multiples

Q: What is a perfect number?

A perfect number equals the sum of its proper divisors. The number 6 is perfect: 1 + 2 + 3 = 6. The number 28 is perfect: 1 + 2 + 4 + 7 + 14 = 28. Perfect numbers are rare — the first four are 6, 28, 496, and 8128.

Q: What are common factors?

Common factors of two numbers m and n are numbers that divide both. For 12 and 18: factors of 12 are {1,2,3,4,6,12}, factors of 18 are {1,2,3,6,9,18}, common factors are {1,2,3,6}. The largest common factor is the GCD.

Q: What are common multiples?

Common multiples of m and n are numbers divisible by both. For 4 and 6: common multiples are 12, 24, 36, 48... The smallest positive common multiple is the LCM. Every common multiple is a multiple of the LCM.

Q: How do you count the number of factors?

Use prime factorization. If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, the number of factors is (a₁+1)(a₂+1)...(aₖ+1). For 72 = 2³ × 3²: (3+1)(2+1) = 12 factors. Each exponent plus one represents choices for that prime.

Q: Why do perfect squares have an odd number of factors?

Factors come in pairs (a, n/a) that multiply to n. For perfect squares, the square root pairs with itself (√n, √n), counting as one factor instead of two. All other pairs contribute two factors, so the total is odd.

Summary: A Toolkit for Divisor Structure

The Two Sides of Divisibility

Every divisibility relationship involves two numbers playing opposite roles. When

3 \mid 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.

Key Terms

Divisor (Factor)— the central concept on this page; an integer that divides another exactly

Multiple— the counterpart to a factor, produced by multiplying out

Prime Number— an integer whose only factors are 1 and itself

Composite Number— an integer with additional factors beyond 1 and itself

Prime Factorization— decomposition into primes, used here to count divisors

See All Arithmetic Definitions →

Factors (Divisors)

A factor of

n

is any integer

a

such that

a \mid n

— meaning

n = a \cdot 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

\sqrt{n}

. If

a

divides

n

, both

a

and

\frac{n}{a}

are factors — a pair produced from a single test. Stopping at

\sqrt{n}

is sufficient because if

a > \sqrt{n}

, then

\frac{n}{a} < \sqrt{n}

, and that smaller partner would already have been found.

For

n = 36

, test integers from

1

6

(since

\sqrt{36} = 6

1 \mid 36

: pair

(1, 36)

2 \mid 36

: pair

(2, 18)

3 \mid 36

: pair

(3, 12)

4 \mid 36

: pair

(4, 9)

5 \nmid 36

: skip.

6 \mid 36

: pair

(6, 6)

.

The complete factor set is

\{1, 2, 3, 4, 6, 9, 12, 18, 36\}

— nine factors, all discovered by testing only six candidates.

For large numbers, divisibility rules speed up the testing. Checking whether

2

3

, or

5

divide

n

takes a glance at the digits, not a full division.

Factor Pairs

Every factor

a

n

has a partner:

\frac{n}{a}

. Together they satisfy

a \cdot \frac{n}{a} = 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 \mid b

— meaning

b = a \cdot 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, \ldots

— an infinite sequence with no largest member. Every positive integer generates infinitely many multiples.

Zero is a multiple of every number, since

0 = a \cdot 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 \cdot q + r

, where

n \cdot 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, \ldots

— 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.

Aspect	Factors of n	Multiples of n
Definition	integers a such that a ∣ n	integers b such that n ∣ b
Direction	downward — each is ≤ n	upward — each positive multiple is ≥ n
Cardinality	finite	infinite
Example for n = 12	{1, 2, 3, 4, 6, 12}	12, 24, 36, 48, 60, …

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.

Common Multiples

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, \ldots

The multiples of

6

are

6, 12, 18, 24, 30, 36, \ldots

The common multiples are

12, 24, 36, 48, \ldots

— 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 \cdot 12

36 = 3 \cdot 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.

Counting Factors

For small numbers, counting factors by listing them is practical. For larger numbers, prime factorization provides a formula.

If

n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}

, the number of positive divisors of

n

is:

(a_1 + 1)(a_2 + 1) \cdots (a_k + 1)

Each prime factor

p_i

can appear in a divisor

0, 1, 2, \ldots, a_i

times — that is

a_i + 1

independent choices. The total number of divisors is the product of all these choices.

The number

72 = 2^3 \cdot 3^2

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 = 2^2 \cdot 5^2

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 = 2^2 \cdot 3 \cdot 5

has

12

factors, while

64 = 2^6

has only

7

Sum of Factors

The sum of all positive divisors of

n

is denoted

\sigma(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

\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28

.

For

n = 6

: the divisors are

1, 2, 3, 6

, and

\sigma(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

\sigma(n)

and

n

classifies every positive integer. When

\sigma(n) - n = n

, the number is perfect. When

\sigma(n) - n > n

, it is abundant. When

\sigma(n) - n < n

, it is deficient. Most numbers are deficient; abundance and perfection are the exceptions.

Classification	Sum-of-proper-divisors test	In σ-notation	Examples
Perfect	sum of proper divisors equals n	σ(n) = 2n	6, 28, 496, 8128 (extremely rare)
Abundant	sum of proper divisors exceeds n	σ(n) > 2n	12 (proper-sum = 16), 18, 20, 24, …
Deficient	sum of proper divisors falls short of n	σ(n) < 2n	every prime; 10 (proper-sum = 8); most numbers

Worked Examples

Find all factors of

48

. Test from

1

\sqrt{48} \approx 6.9

1 \mid 48

(pair

1, 48

2 \mid 48

(pair

2, 24

3 \mid 48

(pair

3, 16

4 \mid 48

(pair

4, 12

5 \nmid 48

6 \mid 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 = 2^2 \cdot 3^2 \cdot 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.

Summary: A Toolkit for Divisor Structure

Every question this page addresses about a number

n

— what its factors are, how many it has, how it compares to similar numbers, what it shares with another integer — reduces to one of a small set of recurring tools. The table below collects those tools in one place, paired with a worked example so each row stands as both a reference and a reminder of how to apply it.

Question about n	Tool or formula	Worked example
Find every factor	pair-search from 1 to √n: each a with a ∣ n gives the pair (a, n / a)	n = 36: test 1–6 → pairs (1,36), (2,18), (3,12), (4,9), (6,6) → 9 factors
Count the factors	factor n = p₁^a₁ · … · p_k^a_k; the number of divisors is (a₁+1)(a₂+1)…(a_k+1)	72 = 2³ · 3² → (3+1)(2+1) = 12 factors
Classify n (perfect / abundant / deficient)	sum the proper divisors and compare to n	6: 1 + 2 + 3 = 6 → perfect; 12: 1+2+3+4+6 = 16 > 12 → abundant
Find common factors with m	intersect the factor sets; the largest is gcd(m, n)	12 ∩ 18 in factors → {1, 2, 3, 6}; gcd = 6
Find common multiples with m	the smallest is lcm(m, n); every common multiple is a multiple of it	4 and 6: lcm = 12; common multiples are 12, 24, 36, 48, …