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Sabse Bada

Class 7 Maths · Chapter 11: Saajha Aadhaar Dhoondhna · NCERT Ganita Prakash section P2 3.1

Seekho

Meher ka kamra 12 ft × 16 ft hai. Use barabar varg tiles chahiye, poore feet, kam se kam. Bhuja 12 (factors 1, 2, 3, 4, 6, 12) aur 16 (1, 2, 4, 8, 16) dono ko baante. Common factors 1, 2, 4. Kam tiles matlab sabse badi: 4 ft, to 3 × 4 = 12 tiles.

HCF(12, 16) = 4

Sabse bada common factor Highest Common Factor (HCF) hai, ise Greatest Common Divisor (GCD) bhi kehte hain. 84 kg aur 108 kg chaawal barabar, kam se kam thailon mein: common factors 1, 2, 3, 4, 6, 12, to 12 kg ke thaile. Aur 0 se sabse lambi barabar chhalaang jo 14 aur 30 dono par utre: HCF = 2.

HCF(84, 108) = 12

Factors ki list dheemi aur galti wali. Prime factorisation madad karta hai: primes se bhaag dete raho. 105 → 3 × 35 → 3 × 5 × 7. Har number ka ek hi prime factorisation (sirf kram badal sakta): 90 = 3 × 30 = 3 × 2 × 15 = 2 × 3 × 3 × 5. Har factor iska hissa hai: 840 = 2 × 2 × 2 × 3 × 5 × 7 ka factor 2 × 2 × 7 = 28, kyunki 840 = 28 × 30.

90 = 2 × 3 × 3 × 5

Primes se HCF: dono mein common primes lo, har ek kam se kam jitni baar aaye. 30 = 2 × 3 × 5 aur 72 = 2 × 2 × 2 × 3 × 3 mein ek 2 aur ek 3 saajha: HCF = 6. 225 = 3 × 3 × 5 × 5 aur 750 = 2 × 3 × 5 × 5 × 5: HCF = 3 × 5 × 5 = 75. Koi prime saajha na ho (96 aur 275) to HCF 1. ("Bade number ka factorisation lamba" jaisa daava conjecture hai: 121 = 11 × 11 ulta-udaharan.)

HCF(30, 72) = 2 × 3 = 6

Explore: common factors.

  1. 12 aur 16 ke har common factor par tap karo.

Yaad rakho

  • HCF = sabse bada common factor; "sabse bada barabar tukda" sawaalon ka jawaab.
  • Prime factorisation se: common primes, har ek kam se kam baar.

Quick check

45 aur 75 ka HCF?

  1. 15
  2. 5
  3. 3
  4. 225
Answer

15. 45 = 3 × 3 × 5, 75 = 3 × 5 × 5: common 3 × 5 = 15.

Socho

72 = 6 × 12 aur 144 = 8 × 18. Kya iska matlab sirf 1 common factor hai?

Ek answer

Nahi. Ye prime factorisation nahi. 72 = 2³ × 3² aur 144 = 2⁴ × 3², to HCF 2³ × 3² = 72 (72, 144 ko baant'ta hai).

Practice

Teen levels (easy, medium, hard) ke naye sawaalon se practice karo, kuch hands-on, hints aur hal ke saath. Topic poora karne ke liye pichhle 5 mein se 4 sahi karo.