Prime factorisation kaise karein

Number ko primes mein todna

56 lo. Ye composite hai: 56 = 4 × 14. Ab 14 bhi composite hai, 14 = 2 × 7, isliye 56 = 4 × 2 × 7. Aur 4 = 2 × 2, isliye 56 = 2 × 2 × 2 × 7. Ab har factor prime hai aur aage nahi toot sakta. Kisi number ko primes ke guna ke roop mein likhna uski prime factorisation hai, aur usme aane waale primes, yahan 2 aur 7, uske prime factors hain.

• 1 se bade har number ki prime factorisation hoti hai.
• Number 1 ki koi nahi, kyunki koi prime use divide nahi karta.
• Prime ki prime factorisation wo khud hai: 7 = 7.

Factor trees

Factor tree har composite number ko do factors mein todta jaata hai jab tak sirf primes na bachein.

36 ko chaahe 2 × 18, 3 × 12, 4 × 9 ya 6 × 6 se shuru karo, aakhir mein hamesha do 2 aur do 3 milte hain. Ye ek kamaal ki baat hai ki har number ki sirf ek prime factorisation hoti hai, factors ke kram ko chhodkar.

Kya kram se farq padta hai?

Nahi. 30 cubes ko ek dabbe ki tarah socho, 2 cube ooncha, 3 cube chauda aur 5 cube lamba: kisi bhi disha mein parat-dar-parat gino, 30 hi aate hain, isliye 2 × 3 × 5 = 3 × 5 × 2 = 5 × 2 × 3. Numbers ko kisi bhi kram mein guna karne par wahi nateeja aata hai, jise aage chalkar commutativity aur associativity ke naam se padhoge. Aadat ke taur par primes badhte kram mein likhte hain: 225 = 3 × 3 × 5 × 5.

Guna ki factorisation

Kisi guna ki factorisation ke liye har hisse ki factorisation nikaalo aur jod do. Kyunki 72 = 12 × 6, 12 = 2 × 2 × 3 aur 6 = 2 × 3, isliye 72 = 2 × 2 × 3 × 2 × 3 = 2 × 2 × 2 × 3 × 3. Har prime 72 mein utni baar aata hai jitni baar 12 aur 6 mein milaakar. Isi tarah 108 × 75 = (2 × 2 × 3 × 3 × 3) × (3 × 5 × 5) = 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5.

Hal kiye udaharan

Number	Prime factorisation
64	2 × 2 × 2 × 2 × 2 × 2
104	2 × 2 × 2 × 13
105	3 × 5 × 7
243	3 × 3 × 3 × 3 × 3
1331	11 × 11 × 11
1000	2 × 2 × 2 × 5 × 5 × 5

Teen alag prime factors waala sabse chhota number 2 × 3 × 5 = 30 hai, aur chaar waala 2 × 3 × 5 × 7 = 210.

Prime factorisation ka use

Co-prime jaanchne ke liye: do numbers co-prime hain agar koi prime dono factorisations mein na aaye. 80 = 2 × 2 × 2 × 2 × 5 aur 63 = 3 × 3 × 7 mein koi common nahi, isliye wo co-prime hain.

Divisibility jaanchne ke liye: ek number doosre ko divide karta hai agar uski prime factorisation doosre mein shaamil ho.

• Kya 168, 12 se divisible hai? 168 = 2 × 2 × 2 × 3 × 7 mein 12 = 2 × 2 × 3 shaamil hai, isliye haan: 168 = 12 × 14.
• Kya 75, 21 se divisible hai? 21 = 3 × 7, par 7, 75 = 3 × 5 × 5 ka prime factor nahi, isliye nahi.
• Kya 42, 12 se divisible hai? 12 ko do 2 chahiye, par 42 = 2 × 3 × 7 mein sirf ek hai, isliye nahi.

Do numbers ka koi factor common hai ya nahi, iske liye padho co-prime numbers; jaldi ke tests ke liye padho divisibility ke rules. Textbook ke sawaalon ke liye dekho Chapter 5 ke NCERT solutions; poore chapter ke liye Chapter 5 ke notes.