Factors of 11
11 is one of those numbers that looks ordinary but behaves in a special way. It is prime — meaning nothing divides into it cleanly except 1 and itself. On this page you will learn exactly what that means, how to prove it, and why 11 comes with one of the cleverest divisibility rules in all of elementary math.
Quick Answer
The factors of 11 are: 1 and 11
Total factors: 2 | Type: Prime | Factor pairs: 1
What is a Factor?
A factor of a number is any whole number that divides into it perfectly — with zero left over. Think of it as sharing equally. If you have 11 marbles and you try to split them into equal groups, the only options that work are:
- — 1 group of 11 marbles
- — 11 groups of 1 marble
Every other split — 2 groups, 3 groups, 4 groups — leaves at least one marble left over. That is why 11 has only two factors.
All Factors of 11
Only two factors. This alone tells you 11 is prime.
How to Find the Factors of 11 (Step by Step)
The method is straightforward: divide 11 by every whole number from 1 up to 11. If the remainder is 0, that number is a factor.
| Division | Result | Remainder | Factor? |
|---|---|---|---|
| 11 ÷ 1 | 11 | 0 | Yes |
| 11 ÷ 2 | 5.5 | 1 | No |
| 11 ÷ 3 | 3.67 | 2 | No |
| 11 ÷ 4 | 2.75 | 3 | No |
| 11 ÷ 5 | 2.2 | 1 | No |
| 11 ÷ 6 | 1.83 | 5 | No |
| 11 ÷ 7 | 1.57 | 4 | No |
| 11 ÷ 8 | 1.38 | 3 | No |
| 11 ÷ 9 | 1.22 | 2 | No |
| 11 ÷ 10 | 1.1 | 1 | No |
| 11 ÷ 11 | 1 | 0 | Yes |
Nine consecutive failures. That is exactly what makes 11 prime.
Factor Pairs of 11
A factor pair is two numbers that multiply together to give 11. With only two factors, there is exactly one pair.
| Factor 1 | Factor 2 | Product |
|---|---|---|
| 1 | 11 | 1 × 11 = 11 |
Prime Factorization of 11
Prime factorization means breaking a number down into a product of prime numbers. For most numbers, this takes several steps. For 11, the work is already done.
Start with 11. Is it divisible by any prime smaller than itself?
- 11 ÷ 2 = 5 remainder 1 — no
- 11 ÷ 3 = 3 remainder 2 — no
- 11 ÷ 5 = 2 remainder 1 — no
- 11 ÷ 7 = 1 remainder 4 — no
We only need to check primes up to √11 ≈ 3.32. Since 2 and 3 both fail, 11 is prime.
The exponent 1 means 11 appears once — and only once. There is nothing simpler to break it into.
Why 11 is a Prime Number
A prime number has exactly two factors: 1 and itself. To confirm 11 is prime, you only need to check divisibility by primes up to its square root (√11 ≈ 3.32). That means checking just 2 and 3.
| Test | Calculation | Verdict |
|---|---|---|
| 11 ÷ 2 | 5 r 1 | Not divisible |
| 11 ÷ 3 | 3 r 2 | Not divisible |
Both fail. No further checks needed. 11 is confirmed prime.
Divisibility Rule for 11
This is one of the most interesting divisibility rules in elementary math. Here it is:
The alternating digit sum rule:
Starting from the right, add every other digit, then subtract the remaining digits. If the result is 0 or divisible by 11, the whole number is divisible by 11.
Worked examples:
| Number | Alternating sum | Divisible by 11? |
|---|---|---|
| 121 | 1 − 2 + 1 = 0 | Yes |
| 132 | 1 − 3 + 2 = 0 | Yes |
| 253 | 2 − 5 + 3 = 0 | Yes |
| 1001 | 1 − 0 + 0 − 1 = 0 | Yes |
| 145 | 1 − 4 + 5 = 2 | No |
| 200 | 2 − 0 + 0 = 2 | No |
Multiples of 11
Multiples are what you get when you multiply 11 by 1, 2, 3, and so on. They go outward from 11. Notice the two-digit multiples — they are all repeating digit pairs, which is part of why the alternating sum rule works so neatly.
The first nine multiples of 11 all have two identical digits — 11, 22, 33 … 99. That pattern breaks at 110.
Where Does 11 Appear in Real Life?
Football team
A standard football (soccer) team has 11 players on the field.
Sides of a hendecagon
A polygon with 11 sides is called a hendecagon — 11 equal sides, all connected.
Apollo 11
The Apollo 11 mission was the first to land humans on the Moon in 1969.
11 in music
Spinal Tap fans know it — when one amp goes to 11, it goes one louder.
Canadian dollar
The Canadian $1 coin (loonie) is an 11-sided polygon.
Clock faces
An analogue clock shows 11 hour numbers before cycling back to 12.
Factors vs Multiples of 11
Students mix these up constantly. The key difference: factors go into 11, multiples come out of 11.
| Property | Factors of 11 | Multiples of 11 |
|---|---|---|
| Direction | Divide into 11 | Multiply out of 11 |
| Count | Finite (only 2) | Infinite |
| Examples | 1, 11 | 11, 22, 33, 44 … |
| Always includes | 1 and 11 | 11 itself |
How 11 Compares to Nearby Numbers
| Number | Factors | Count | Type |
|---|---|---|---|
| 8 | 1, 2, 4, 8 | 4 | Composite |
| 9 | 1, 3, 9 | 3 | Composite |
| 10 | 1, 2, 5, 10 | 4 | Composite |
| 11 | 1, 11 | 2 | Prime |
| 12 | 1, 2, 3, 4, 6, 12 | 6 | Composite |
| 13 | 1, 13 | 2 | Prime |
Common Mistakes to Avoid
Assuming 11 is composite
11 looks like a two-digit number built from 1s, but that does not make it composite. The number of digits has nothing to do with being prime.
Thinking 11 ÷ 11 is not valid
Every number divides by itself. 11 ÷ 11 = 1 exactly, so 11 is always a factor of itself.
Confusing factors with digits
11 contains the digit 1 twice, but 1 is still only one factor, not two. We list 1 once in the factor list.
Forgetting the square root shortcut
You only need to test primes up to √11 ≈ 3.32 to confirm primality. Many students keep dividing all the way to 10 unnecessarily.
Quick Reference Summary
| All factors | 1, 11 |
| Number of factors | 2 |
| Factor pairs | (1, 11) |
| Prime factorization | 11¹ |
| Type | Prime |
| Perfect square? | No |
| Divisibility rule | Alternating digit sum = 0 or multiple of 11 |