HCF aur LCM ka sambandh

Niyam

Koi bhi do numbers lo, unka HCF aur LCM nikaalo, aur dono jawabon ko guna karo. Hamesha un numbers ka guna milta hai:

HCF × LCM = pehla number × doosra number

105 = 3 × 5 × 7 aur 95 = 5 × 19 ka LCM 3 × 5 × 7 × 19 = 1995 aur HCF 5 hai; 1995 × 5 = 9975 = 105 × 95. 45 aur 105 ka LCM 315 aur HCF 15 hai, aur 315 × 15 = 4725 = 45 × 105.

Ye kyun chalta hai

Dono numbers ke primes ko do ek-doosre pe chadhe gol gheron mein rakho. Saajha primes beech ke hisse mein aate hain aur baaki bahar. HCF beech ka hissa hai. LCM sab kuch hai: beech ka hissa ek baar, aur wo primes jo sirf ek number mein hain. Isliye HCF aur LCM milkar beech ka hissa do baar aur baaki har prime ek baar rakhte hain, theek utna jitna dono numbers milkar rakhte hain. 24 = 2 × 2 × 2 × 3 aur 36 = 2 × 2 × 3 × 3 ke liye HCF 2 × 2 × 3 = 12, LCM 2 × 2 × 2 × 3 × 3 = 72, aur 12 × 72 = 864 = 24 × 36.

Is niyam se do kaam ki baatein nikalti hain. LCM kabhi guna se zyaada nahi hota, aur theek guna ke barabar tab hota hai jab HCF 1 ho. Aur agar chaar mein se teen values pata hon, to chauthi nikal sakti hai: agar do numbers ka HCF 4 aur LCM 48 hai aur ek number 12 hai, to doosra 4 × 48 ÷ 12 = 16 hai.

Ye niyam do numbers ke liye hai. 2, 4 aur 8 ka HCF 2 aur LCM 8 hai, par 2 × 8 = 16, 2 × 4 × 8 = 64 ke barabar nahi.

Patterns

• Ek number doosre ka factor (n aur kn): HCF n aur LCM kn. 6 aur 18 ke liye ye 6 aur 18 hain.
• Lagaatar numbers jaise 8 aur 9: HCF 1 aur LCM 72, unka guna.
• Lagaatar even numbers jaise 10 aur 12: HCF 2, isliye LCM guna ka aadha, 60.
• Lagaatar odd numbers jaise 15 aur 17: HCF 1, LCM 255.
• Dono numbers dugne: HCF aur LCM bhi dugne.

Aasaan ladder tareeka

Dono ek saath nikaalne ke liye do numbers ko agal-bagal likho aur dono ko ek saajha prime se bhaag do, bhaagfal neeche likho. Tab tak dohraao jab tak bache hue do numbers mein koi prime saajha na rahe.

Divisors ka guna HCF hai: 84 aur 180 ke liye 2 × 2 × 3 = 12. Divisors ko neeche bache numbers se guna karne pe LCM milta hai: 12 × 7 × 15 = 1260. Ye isliye chalta hai kyunki har number HCF guna apna bacha hua hissa hai, aur bache hisson mein kuch saajha nahi, isliye kuch hataaya nahi ja sakta. Sirf prime nahi, koi bhi dikhne waala common factor le sakte ho: 630 aur 770 ko 10 se aur phir 7 se bhaag dekar 9 aur 11 milte hain, isliye HCF 70 aur LCM 70 × 9 × 11 = 6930 hai.

Ye kyun zaroori hai

Ye niyam jawab jaldi jaanchta hai aur gaayab number dhoondh deta hai. Har hissa alag se seekho HCF kaise nikalein? aur LCM kaise nikalein? mein. Textbook ke sawaalon ke jawab Part 2 Chapter 3 ke NCERT solutions mein hain; poore Part 2 Chapter 3 ke notes dekho.