06. Prime, Composite & Co-prime Numbers

Prime, Composite aur Co-prime numbers ka complete concept: definitions, properties, prime factorization link, tests, common traps, SSC CGL level examples & practice questions with answers.

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1) Prime Number (Prime)

Prime number wo natural number (> 1) hota hai jiske exactly 2 factors hote hain: 1 aur number itself.

• Examples: 2, 3, 5, 7, 11, 13, 17, 19, ...
• Smallest prime: 2
• Only even prime: 2

2) Composite Number

Composite number2 se zyada factors hote hain. Means: prime nahi hai.

• Examples: 4 (1,2,4), 6 (1,2,3,6), 8, 9, 10, 12, ...

3) 1 ka special case (most asked)

1 na prime hai na composite. Prime ke liye 2 factors must hain, but 1 ke sirf 1 factor hota hai (1).

4) Prime check ka fast idea (√n rule)

Kisi number n ko prime check karne ke liye sirf 2 se लेकर √n tak divisibility check karna enough hota hai. Agar √n tak koi divisor nahi mila ⇒ n prime.

Example: 29 prime?

• √29 ≈ 5.3 ⇒ check 2,3,5
• 29 not divisible by 2,3,5 ⇒ prime

5) Co-prime Numbers (Relative Primes)

Do numbers co-prime tab hote hain jab unka HCF/GCD = 1 ho. Co-prime hone ke liye numbers ka prime hona zaroori nahi.

• Example: 8 and 15 (HCF=1) ⇒ co-prime
• Example: 14 and 25 (HCF=1) ⇒ co-prime
• Example: 12 and 25 (HCF=1) ⇒ co-prime

Very important points

• Consecutive numbers always co-prime: (n, n+1) ⇒ HCF = 1
• If two numbers are co-prime, then unka common factor sirf 1 hota hai.

6) Prime Factorization link (quick)

Two numbers co-prime tab hote hain jab unke prime factors me koi overlap na ho.

• 8 = 2³, 15 = 3×5 ⇒ common prime factor none ⇒ co-prime
• 12 = 2²×3, 18 = 2×3² ⇒ common primes 2,3 ⇒ not co-prime

7) Some SSC quick facts

• 2 aur 3 ke baad saare primes odd hote hain.
• 2 se bada koi even number prime nahi hota.
• Composite numbers: 4 se start hote hain.
• Twin primes: primes ka pair jisme difference 2 ho (3,5), (11,13), (17,19) etc.

8) Common traps

• 1 ko prime mat samajhna.
• Co-prime ka matlab “both prime” nahi hota.
• Even numbers me sirf 2 prime hai, baaki sab composite.

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9) Practice (SSC CGL) + Answers

1. Prime/composite/none: 1, 2, 9, 17
2. 29 prime hai ya composite?
3. Co-prime pair identify karo: (14,21), (8,15), (12,25)
4. Which is not a prime? 31, 37, 39, 41
5. True/False: Consecutive numbers always co-prime.
6. HCF 1 hai: 18 aur 35 — co-prime? (Yes/No)

Show Answers

1. 1: none; 2: prime; 9: composite; 17: prime
2. Prime (√29 ≈ 5.3; not divisible by 2,3,5)
3. (14,21) not; (8,15) yes; (12,25) yes
4. 39 (3×13)
5. True
6. Yes (18 factors: 2,3; 35 factors: 5,7; common none ⇒ HCF 1)