Lagnever AI English

HCF kaise nikalein?

Short mein jawab

Do ya zyaada numbers ka HCF (highest common factor) wo sabse bada number hai jo sabko poora-poora bhaag de. Common factors ki list banaakar sabse bada lo, ya prime factorisation lagao: sabhi numbers mein common primes ko guna karo, har ek ko utni baar jitni sabse kam baar wo aata hai. 72 aur 120 ka HCF 24 hai.

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Common factors

Kisi number ka factor use poora-poora bhaag deta hai. Do ya zyaada numbers ka common factor un sabko bhaag deta hai. Common factors cheezon ko barabar hisson mein baantne ke sawaalon ka jawab dete hain. Maan lo farsh 12 ft by 16 ft hai aur hum barabar square tiles chahte hain jinki side poore feet mein ho, bina kaate. Side 12 ko bhaag de (chaudai mein fit ho) aur 16 ko bhi (lambai mein fit ho), isliye wo 12 aur 16 ka common factor hogi.

Wahi 12 ft by 16 ft ka kamra 1 ft tiles (192), 2 ft tiles (48) aur 4 ft tiles (12) se
Badi tiles, kam tiles

12 ke factors: 1, 2, 3, 4, 6, 12. 16 ke factors: 1, 2, 4, 8, 16. Common factors: 1, 2 aur 4. Kam se kam tiles ke liye sabse badi, 4 ft, chuno, jisme 3 × 4 = 12 tiles lagti hain. Side ko fraction mein lene se bhi fayda nahi: jo side dono lambaiyon mein poori baar fit ho, wo zyaada se zyaada 4 ft ho sakti hai.

Highest common factor

HCF (highest common factor) common factors mein sabse bada hai; iska doosra naam GCD, greatest common divisor, hai. Jab bhi sawaal poochhe ki kai maatraon mein poori tarah fit hone waala sabse bada barabar size kya hai, jawab HCF hota hai. Agar 84 kg aur 108 kg chawal barabar bags mein bhara jaaye, aur har bag mein ek hi khet ka chawal ho, to 84 aur 108 ke common factors 1, 2, 3, 4, 6 aur 12 hain. Sabse kam bags 12 kg ke bags se bante hain: 7 + 9 = 16 bags.

0 se 10 ki jumps 30 aur 50 pe utarti hain; 14 ki jumps 28 aur 42 pe
Dono tak pahunchne waali sabse lambi jump

Jump Jackpot game yahi baat number line pe dikhata hai. 0 se shuru karke koi jump size sirf apne multiples pe utarta hai, isliye dono khazaanon pe utarne ke liye use unki jagahon ka common factor hona chahiye. Sabse lambi aisi jump HCF hai: 30 aur 50 ke khazaanon ke liye 10, aur 28 aur 42 ke liye 14.

Prime factorisation se

Bade numbers ke saare factors ki list banaana dheema hai, aur ek factor chhoot jaaye to jawab galat ho jaata hai. Prime factorisation zyaada bharosemand hai. Kisi number ka har factor uske prime factorisation ka hissa hai, isliye common factor har factorisation ka hissa hona chahiye. Sabse bada saajha hissa HCF hai:

  • sirf wo primes rakho jo sabhi numbers mein hon;
  • har ek ko utni baar lo jitni sabse kam baar wo aata hai.
72 aur 120 ko 2s, 3s aur 5s mein toda gaya: HCF teen 2 aur ek 3 leta hai
72 aur 120 ka HCF = 24

72 = 2 × 2 × 2 × 3 × 3 aur 120 = 2 × 2 × 2 × 3 × 5 mein dono mein teen 2 hain, aur 3 ki kam ginti ek hai; 5 saajha nahi hai. Isliye HCF = 2 × 2 × 2 × 3 = 24. 225 = 3 × 3 × 5 × 5 aur 750 = 2 × 3 × 5 × 5 × 5 ka HCF 3 × 5 × 5 = 75 hai. Teen numbers pe bhi yahi niyam chalta hai: 42 = 2 × 3 × 7, 75 = 3 × 5 × 5 aur 24 = 2 × 2 × 2 × 3 mein sirf ek 3 saajha hai, isliye HCF 3 hai.

Numbers ko poori tarah primes tak todna zaroori hai. 72 = 6 × 12 aur 144 = 8 × 18 likhne se lag sakta hai ki kuch saajha nahi, par primes mein 72 = 2 × 2 × 2 × 3 × 3 aur 144 = 2 × 2 × 2 × 2 × 3 × 3 hain, isliye HCF 72 hai.

Jab HCF 1 ho

Agar do numbers mein koi prime saajha na ho, to unka ek hi common factor 1 hai, aur unhe co-prime kehte hain. Jaise 96 = 2 × 2 × 2 × 2 × 2 × 3 aur 275 = 5 × 5 × 11 ka HCF 1 hai. Do lagaatar numbers, jaise 20 aur 21, hamesha co-prime hote hain, kyunki 1 se bada koi number aise do numbers ko bhaag nahi de sakta jo sirf 1 ki doori pe hon.

Ye kyun zaroori hai

HCF sabse bada barabar hissa batata hai: sabse badi tile, rassi ka sabse lamba tukda, bina kuch bache sabse bada group. Fractions ko chhota karne mein bhi kaam aata hai, kyunki numerator aur denominator ko unke HCF se bhaag dene pe sabse chhota roop milta hai. Aage dekho LCM kaise nikalein? aur HCF aur LCM ka sambandh. Textbook ke sawaalon ke jawab Part 2 Chapter 3 ke NCERT solutions mein hain; poore Part 2 Chapter 3 ke notes dekho.

Practice karo

  1. 12 aur 18 ke common factors hain:

    1. (a) 1, 2, 3, 6
    2. (b) 1, 2, 3, 4, 6
    3. (c) 2, 3
    4. (d) 1, 6, 36
    Jawab dekho

    Jawab: (a) 1, 2, 3, 6

  2. 2 × 2 × 3 × 5 aur 2 × 3 × 3 × 7 ka HCF hai:

    1. (a) 2
    2. (b) 6
    3. (c) 30
    4. (d) 1260
    Jawab dekho

    Jawab: (b) 6

  3. 48 m aur 60 m ki rassiyon ko jitne lambe ho sake utne barabar tukdon mein kaata gaya, bina kuch bache. Har tukda hai:

    1. (a) 6 m
    2. (b) 12 m
    3. (c) 24 m
    4. (d) 4 m
    Jawab dekho

    Jawab: (b) 12 m

  4. Prime factorisation se 84 aur 126 ka HCF nikaalo.

    Jawab dekho

    Jawab: 84 = 2 × 2 × 3 × 7 aur 126 = 2 × 3 × 3 × 7. Common primes, har ek sabse kam baar: 2 × 3 × 7 = 42.

  5. 35 aur 64 ka HCF 1 kyun hai?

    Jawab dekho

    Jawab: 35 = 5 × 7 aur 64 = 2 × 2 × 2 × 2 × 2 × 2 mein koi prime saajha nahi hai, isliye unka ek hi common factor 1 hai.

Students ke sawal

HCF kya hai?

Do ya zyaada numbers ka HCF, yaani highest common factor, wo sabse bada number hai jo un sabka factor ho. 12 aur 16 ke common factors 1, 2 aur 4 hain, isliye HCF 4 hai.

Kya HCF aur GCD ek hi hain?

Haan. Highest common factor (HCF) aur greatest common divisor (GCD) ek hi number ke do naam hain: wo sabse bada number jo diye gaye sabhi numbers ko poora-poora bhaag de.

Teen numbers ka HCF kaise nikaalein?

Teeno ko primes mein todo, sirf wo primes rakho jo har number mein hon, har ek ko sabse kam baar jitna wo aata hai, aur guna karo. 42, 75 aur 24 mein teeno mein sirf 3 hai, isliye HCF 3 hai.

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