Prime & Composite Numbers
A prime has exactly two factors. A composite has more. And 1 is neither — the most common MCQ trap.
A. Key Definitions
Prime Number
A number that has exactly two factors — 1 and itself. Nothing else divides it.
Example
7 is prime → factors are only 1 and 7.
13 is prime → factors are only 1 and 13.
13 is prime → factors are only 1 and 13.
Composite Number
A number that has more than two factors. It can be divided by something other than just 1 and itself.
Example
12 is composite → factors are 1, 2, 3, 4, 6, 12. (six factors)
9 is composite → factors are 1, 3, 9. (three factors)
9 is composite → factors are 1, 3, 9. (three factors)
Neither — the number 1
The number 1 is neither prime nor composite. It has only one factor — itself. Prime needs exactly two factors, so 1 does not qualify.
Example
1 → only factor is 1 itself.
Not prime (needs 2 factors). Not composite (needs more than 2). It is neither.
Not prime (needs 2 factors). Not composite (needs more than 2). It is neither.
Co-prime Numbers
Two numbers are co-prime when their HCF equals 1. They do not share any common factor other than 1.
Example
8 and 9 are co-prime → HCF(8, 9) = 1.
4 and 9 are co-prime → HCF(4, 9) = 1.
4 and 6 are NOT co-prime → HCF(4, 6) = 2.
4 and 9 are co-prime → HCF(4, 9) = 1.
4 and 6 are NOT co-prime → HCF(4, 6) = 2.
⚠ Most tested fact: 1 is neither prime nor composite. This appears in almost every MCQ on this topic. Also — 2 is the only even prime number. Every other even number divides by 2, giving it more than 2 factors.
B. Primes to Know by Heart
These are the prime numbers up to 50. The ones in bold are most tested in MCQs.
Primes 1–30 — Green = Prime, Grey = Composite, Red = Neither
Primes between 1 and 10
There are 4 prime numbers here.
List2, 3, 5, 7
Primes between 11 and 20
There are 4 prime numbers here.
List11, 13, 17, 19
Primes between 21 and 50
There are 7 prime numbers here.
List23, 29, 31, 37, 41, 43, 47
Numbers that look prime but are NOT
These trip up students constantly. Always double-check them.
Common traps
9 = 3 × 3 → composite
21 = 3 × 7 → composite
49 = 7 × 7 → composite
91 = 7 × 13 → composite (biggest MCQ trap!)
21 = 3 × 7 → composite
49 = 7 × 7 → composite
91 = 7 × 13 → composite (biggest MCQ trap!)
Quick tip: Any number ending in 0, 2, 4, 6, 8 (except 2) is not prime — it divides by 2. Any number ending in 5 (except 5 itself) is not prime — it divides by 5.
C. How to Test if a Number is Prime
You do not need to try every number. Only divide by primes up to the square root of the number. If nothing divides it exactly — it is prime.
The Test
Try dividing by: 2, 3, 5, 7 … up to √n
If none divide exactly → prime · If one divides exactly → composite
Example 1 — Is 37 prime?
1Find √37 ≈ 6.1 → only need to test up to 6
237 ÷ 2 = 18.5 → not exact
337 ÷ 3 = 12.3 → not exact
437 ÷ 5 = 7.4 → not exact
5No divisor found up to √37 → 37 is prime
37 is PRIME
Example 2 — Is 91 prime? (Famous MCQ trap)
1√91 ≈ 9.5 → test: 2, 3, 5, 7
291 ÷ 2 = 45.5 → not exact
391 ÷ 3 = 30.3 → not exact
491 ÷ 5 = 18.2 → not exact
591 ÷ 7 = 13 exactly ✓ → 91 = 7 × 13
91 is COMPOSITE — it equals 7 × 13
Example 3 — Is 53 prime?
1√53 ≈ 7.3 → test: 2, 3, 5, 7
253 ÷ 2 = 26.5 ✗ 53 ÷ 3 = 17.6 ✗ 53 ÷ 5 = 10.6 ✗ 53 ÷ 7 = 7.57 ✗
3Nothing divides exactly up to √53
53 is PRIME
⚠ Remember 91: 91 looks prime — it is odd and does not end in 5. But 91 = 7 × 13. This exact number appears in MCQs regularly. Always test up to √n.
Quick MCQ Revision
| Fact | Answer |
|---|---|
| Is 1 prime? | No — has only 1 factor, needs exactly 2 |
| Only even prime | 2 |
| Primes 1–20 | 2, 3, 5, 7, 11, 13, 17, 19 |
| Is 9 prime? | No — 9 = 3 × 3 (composite) |
| Is 91 prime? | No — 91 = 7 × 13 (composite) |
| Co-prime means | HCF of the two numbers = 1 |