05. Factors & Multiples

Factors aur Multiples ka complete concept: definitions, properties, prime factorization, common factors/multiples, divisibility idea, SSC CGL tricks + practice questions with answers.

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1) Factor kya hota hai?

Factor wo number hota hai jo kisi number ko completely divide kare (remainder 0).

• 12 ke factors: 1, 2, 3, 4, 6, 12
• Check: 12 ÷ 3 = 4 (remainder 0) ⇒ 3 is a factor of 12

2) Multiple kya hota hai?

Multiple

• 3 ke multiples: 3, 6, 9, 12, 15, 18, ...
• 12 is a multiple of 3 because 12 = 3×4

3) Factor vs Multiple (one-line difference)

• Factor: number ko divide karta hai
• Multiple: number se multiply karke banta hai

4) Important properties (SSC CGL)

• Har number ka smallest factor 1 hota hai.
• Har number ka greatest factor number itself hota hai.
• Har number ka smallest positive multiple number itself hota hai.
• Multiples infinite hote hain, factors finite hote hain.
• अगर a is a factor of b, then b is a multiple of a.

5) How to list factors fast?

Factors pair me aate hain: agar a × b = n, to a aur b dono n ke factors hain. Isliye factors nikalne ke liye √n tak check karo.

Example: 36

• 1×36
• 2×18
• 3×12
• 4×9
• 6×6

So factors: 1,2,3,4,6,9,12,18,36

6) Prime Factorization (very important base)

Kisi number ko primes ke product me todna prime factorization hota hai. Ye HCF/LCM, divisors count, highest power, etc. me direct use hota hai.

Example: 72 ka prime factorization

72 = 2×36 = 2×2×18 = 2×2×2×9 = 2³ × 3²

7) Common Factors & Common Multiples

Common Factors: jo dono numbers ko divide kare.

• Example: 12 and 18
• Factors(12): 1,2,3,4,6,12
• Factors(18): 1,2,3,6,9,18
• Common factors: 1,2,3,6

Common Multiples: jo dono numbers ke multiples me aate hain.

• Multiples of 4: 4,8,12,16,20,24,...
• Multiples of 6: 6,12,18,24,30,...
• Common multiples: 12,24,36,...

8) SSC favourite: Greatest common factor vs Least common multiple

Common factors me jo sabse bada hota hai = HCF/GCD. Common multiples me jo sabse chhota positive hota hai = LCM. (Detailed pages HCF-LCM topic me aayenge, yaha base clear rakho.)

9) Common traps

• “Factor” aur “multiple” ulta mat karna (exam me common confusion).
• 0 multiple hota hai? — 0 is a multiple of every non-zero integer (because 0 = n×0), but factor concept me 0 tricky: 0 kisi non-zero number ka factor nahi hota.
• Factors nikalte time √n se aage check karna unnecessary hota hai.

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10) Practice Questions (SSC CGL) + Answers

1. 24 ke factors likho.
2. 5 ke first 6 multiples likho.
3. 36 aur 54 ke common factors ka set likho.
4. 60 ka prime factorization likho.
5. True/False: 1 is a multiple of every number.
6. Which number has more factors: 16 or 18?

Show Answers

1. 1, 2, 3, 4, 6, 8, 12, 24
2. 5, 10, 15, 20, 25, 30
3. Factors(36): 1,2,3,4,6,9,12,18,36; Factors(54): 1,2,3,6,9,18,27,54; Common: 1,2,3,6,9,18
4. 60 = 2² × 3 × 5
5. False (1 is multiple of only 1)
6. 16 has factors 1,2,4,8,16 (5); 18 has 1,2,3,6,9,18 (6) ⇒ 18 has more