Types of numbers — prime, composite, perfect, twin primes, co-prime explained

Question

Is 51 a prime number or composite? Find all prime numbers between 1 and 30. Are 8 and 15 co-prime?

(CBSE Class 6 — Whole Numbers / Playing with Numbers)

Number Type Classification

flowchart TD
    A["Natural Numbers"] --> B{"How many factors?"}
    B -->|"Exactly 1 factor"| C["1 (neither prime nor composite)"]
    B -->|"Exactly 2 factors"| D["Prime Number"]
    B -->|"More than 2 factors"| E["Composite Number"]
    D --> F["Examples: 2, 3, 5, 7, 11, 13..."]
    E --> G["Examples: 4, 6, 8, 9, 10, 12..."]
    D --> H["2 is the only even prime"]
    A --> I["Special Types"]
    I --> I1["Twin Primes: differ by 2"]
    I --> I2["Co-prime: HCF = 1"]
    I --> I3["Perfect Numbers: sum of factors = number"]

Solution — Step by Step

To check if 51 is prime, we test divisibility by primes up to $\sqrt{51} \approx 7.1$ .

Divisible by 2? No (51 is odd)
Divisible by 3? $5 + 1 = 6$ , which is divisible by 3. So yes: $51 = 3 \times 17$
Since 51 has factors other than 1 and itself, 51 is composite.

Many students think 51 is prime because it “looks” prime. Always check.

Using the Sieve of Eratosthenes — cross out multiples of 2, 3, 5 (primes up to $\sqrt{30} \approx 5.5$ ):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 — that is 10 prime numbers.

Note: 1 is NOT prime (it has only one factor, not two). 2 is the only even prime number.

Two numbers are co-prime if their HCF (Highest Common Factor) is 1.

Factors of 8: 1, 2, 4, 8

Factors of 15: 1, 3, 5, 15

Common factor: only 1

\boxed{\text{HCF}(8, 15) = 1, \text{ so 8 and 15 are co-prime.}}

Note: co-prime numbers do not have to be prime themselves. 8 and 15 are both composite, but they are still co-prime.

Why This Works

The fundamental theorem of arithmetic states that every number greater than 1 can be expressed as a product of primes in exactly one way. Prime numbers are the “building blocks” — they cannot be broken down further. Composite numbers are products of primes. This classification helps us understand divisibility, HCF, LCM, and many other concepts.

Alternative Method — Divisibility Rules Quick Reference

Divisible by	Rule
2	Last digit is even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Ends in 0 or 5
6	Divisible by both 2 and 3
9	Sum of digits divisible by 9
11	Alternating sum of digits divisible by 11

For CBSE Class 6, the most asked question types are: “Is X prime?”, “Find twin primes between A and B”, and “Are X and Y co-prime?” Twin primes are prime pairs that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31). Memorise the first few pairs.

Common Mistake

Two big errors: (1) calling 1 a prime number — it is not, because a prime must have exactly two distinct factors (1 and itself), while 1 has only one factor. (2) Assuming co-prime means both numbers must be prime. Co-prime only means their HCF is 1. Even 4 and 9 are co-prime (HCF = 1) despite both being composite.