Lagnever AI English

Prime aur composite numbers

Short mein jawab

Prime number ke theek do factors hote hain, 1 aur khud, jaise 2, 3, 5, 7 aur 11. Composite number ke do se zyada factors hote hain, jaise 4 aur 9. Number 1 ka sirf ek factor hai, isliye wo na prime hai na composite. 100 tak 25 primes hain, aur 2 akela even prime hai.

Koi sawal atak gaya? Photo lo, Lagnever AI step-by-step samjhayega. 7 din ka trial sirf ₹2. Google Play se app lo

Anjeer ke aayat

Guna ek dabbe mein 12 anjeer pack karna chaahti hai aur Anshu 7. Guna 12 anjeer ko kai tarah ke aayaton mein laga sakti hai: 12 ki 1 kataar, 6-6 ki 2 kataarein, ya 4-4 ki 3 kataarein. Har baar kataarein × column = 12, isliye kataaron aur columns ki ginti 12 ke factors hain. Anshu sirf ek aayat bana sakta hai, 7 ki 1 kataar, kyunki 7 ke sirf do factors hain, 1 aur 7.

Barah anjeer teen alag aayat banaate hain, 1 × 12, 2 × 6 aur 3 × 4, kyunki 12 composite hai; saat anjeer sirf 1 × 7 banaate hain, kyunki 7 prime hai
12 anjeer aur 7 anjeer

Prime, composite aur 1

  • Prime number ke theek do factors hote hain: 1 aur wo number khud. Pehle primes 2, 3, 5, 7, 11, 13, 17 aur 19 hain.
  • Composite number ke do se zyada factors hote hain, yaani 1 aur khud ke alawa kam se kam ek aur factor. Pehle composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 aur 20 hain.
  • Number 1 ka sirf ek factor hai, isliye wo na prime hai na composite.

21 se 30 tak sirf do primes hain, 23 aur 29; baaki aath numbers composite hain.

Sieve of Eratosthenes

Lagbhag 2200 saal pehle ke Greek ganitagya Eratosthenes ne primes ki list banaane ka ek badhiya tareeka diya:

  1. 1 ko kaato, kyunki wo na prime hai na composite.
  2. 2 par gola lagaao aur uske baad ke uske saare multiples kaato: 4, 6, 8, …
  3. Agla bina kata number 3 hai. Us par gola lagaao aur uske multiples kaato: 6, 9, 12, …
  4. 5 par gola lagaakar uske multiples kaato, phir 7 ke saath yahi karo, aur tab tak chalte raho jab tak har number par gola ya kaat na lag jaaye.
1 se 100 tak Sieve of Eratosthenes: 25 primes par gola, aur composite numbers 2, 3, 5 ya 7 ke multiples ke roop mein kate hue
Sieve of Eratosthenes

Ye kaam kyun karta hai: har kata hua number kisi chhote prime ka multiple hai, isliye uska 1 aur khud ke alawa bhi ek factor hai, yaani wo composite hai. Jo number kabhi nahi kat-ta, uska koi chhota prime factor nahi, isliye uske factors sirf 1 aur khud hain, yaani wo prime hai. 100 tak ke 25 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 aur 97.

Primes ke patterns

  • 2 akela even prime hai, kyunki baaki har even number ka factor 2 hai.
  • Do lagaataar primes ka sabse chhota farq 1 hai (2 aur 3 ke beech); 100 tak sabse bada farq 8 hai (89 aur 97 ke beech).
  • 90 se 96 tak saat lagaataar composite numbers hain.
  • Twin primes ka farq 2 hota hai: 3 aur 5, 5 aur 7, 11 aur 13, 17 aur 19, 29 aur 31, 41 aur 43, 59 aur 61, aur 71 aur 73.
  • Kuch primes ulte jode mein aate hain: 13 aur 31, 17 aur 71, 37 aur 73, 79 aur 97.
  • Kisi prime ka ikaai ank 4 nahi hota, kyunki aisa number even aur 2 se bada hota hai.
  • 3 ke baad har prime ke agle number even hota hai, isliye wo composite hai.

Primes kitne hain?

Prime numbers ko saare poorn sankhyaaon (whole numbers) ki neev kaha jaata hai, aur ganitagya 2000 se zyada saal se inhe padh rahe hain. Book ka mazedaar tathya: ab tak likha gaya sabse bada prime lagbhag 6500 panno mein aayega, isliye wo sirf computer par hai. Book ye bhi poochhti hai ki kya primes ki list kabhi khatm hoti hai. (Extra: nahi hoti: Greek ganitagya Euclid ne saabit kiya ki sabse bada prime koi nahi.)

Primes se doosre numbers kaise bante hain, ye jaanne ke liye padho prime factorisation kaise karein. Textbook ke sawaalon ke liye dekho Chapter 5 ke NCERT solutions; poore chapter ke liye Chapter 5 ke notes.

Practice karo

  1. Inme se prime number kaunsa hai?

    1. (a) 21
    2. (b) 27
    3. (c) 29
    4. (d) 33
    Jawab dekho

    Jawab: (c) 29

  2. 21 se 30 tak kitne primes hain?

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

    Jawab: (b) 2

  3. Kaunsa number na prime hai na composite?

    1. (a) 0
    2. (b) 1
    3. (c) 2
    4. (d) 3
    Jawab dekho

    Jawab: (b) 1

  4. 100 se neeche saat lagaataar composite numbers dhoondho.

    Jawab dekho

    Jawab: 90, 91, 92, 93, 94, 95 aur 96. Ye primes 89 aur 97 ke beech hain: 91 = 7 × 13, 93 = 3 × 31, 95 = 5 × 19, aur baaki even hain.

  5. Kya 2 ke alawa koi even prime hai? Kyun?

    Jawab dekho

    Jawab: Nahi. 2 se bade har even number ka 1 aur khud ke alawa 2 bhi factor hai, isliye uske do se zyada factors hain aur wo composite hai.

Students ke sawal

Prime number kya hota hai?

Wo number jiske theek do factors hon, 1 aur wo number khud. Pehle primes 2, 3, 5, 7, 11, 13, 17 aur 19 hain.

1 prime number kyun nahi hai?

Prime ke theek do factors hone chahiye. Number 1 ka sirf ek factor hai, khud, isliye wo na prime hai na composite.

Twin primes kya hote hain?

Primes ke jode jinka farq 2 ho. 100 tak ye hain: 3 aur 5, 5 aur 7, 11 aur 13, 17 aur 19, 29 aur 31, 41 aur 43, 59 aur 61, aur 71 aur 73.

Koi sawal atak gaya?

Photo lo, Lagnever AI step-by-step samjhayega. 7 din ka trial sirf ₹2.

Google Play se app lo Browser mein kholo

Parents ke liye: trial ke baad plan har mahine UPI ya card autopay se renew hota hai. App mein kabhi bhi cancel kar sakte hain.