Divisibility ke rules

Divisibility tests kyun?

8560 jaise bade number ke factors dhoondhne ke liye har chhote number se lamba bhaag karna dheema hai. Kuch divisors ke liye aakhri ankon par ek nazar kaafi hai. Ye poochhna ki kya 8560, 10 se divisible hai, wahi hai jo ye poochhna ki kya 10, 8560 ka factor hai.

10, 5 aur 2 se: aakhri ank

• 10 se: 10 ke multiples 10, 20, 30, … sab 0 par khatm hote hain. Isliye 8560, 10 se divisible hai, par 125 nahi.
• 5 se: 5 ke multiples 0 ya 5 par khatm hote hain. 399 se chhota sabse bada number jo 5 se divisible ho, 395 hai.
• 2 se: 2 ke multiples 0, 2, 4, 6 ya 8 par khatm hote hain, isliye 682 aur 8560, 2 se divisible hain. 399 aur 411 ke beech 2 ke multiples 400, 402, 404, 406, 408 aur 410 hain.

Ye isliye chalte hain kyunki har number 10 ka koi multiple jama uska aakhri ank hai, aur 10, 2 aur 5 dono ka multiple hai.

4 se: aakhri do ank

4 ke liye aakhri ank kaafi nahi: 12 aur 22 dono 2 par khatm hote hain, par 12, 4 ka multiple hai aur 22 nahi. Iske bajaay aakhri do ank dekho. 330 aur 340 ke beech 4 ke multiples 332 aur 336 hain; 1730 aur 1740 ke beech 1732 aur 1736; aur 2030 aur 2040 ke beech 2032 aur 2036. Aakhri hisse dohraate hain, kyunki 100, 4 ka multiple hai, isliye saikde aur hazaar shesh nahi badalte. Isliye 8536, 4 se divisible hai, kyunki 36 hai.

8 se: aakhri teen ank

8 ke liye aakhri do ank kaafi nahi: 200, 8 ka multiple hai par 100 nahi. Kyunki 1000, 8 ka multiple hai, isliye aakhri teen ank faisla karte hain. 120 aur 140 ke beech 8 ke multiples 128 aur 136 hain; 1120 aur 1140 ke beech 1128 aur 1136; 3120 aur 3140 ke beech 3128 aur 3136. Isliye 8560, 8 se divisible hai, kyunki 560 = 8 × 70. Iske aakhri do ank 68 karne par 8568 milta hai, jo bhi 8 ka multiple hai.

Tests ko milaana

Agar koi number 8 se divisible hai, to 4 aur 2 se bhi hai. Agar wo 2 aur 5 dono se divisible hai, to 10 se bhi hai. Isliye 2, 4, 5, 8 aur 10 ek saath jaanchne ke liye 8 aur 5 jaanchna kaafi hai. Isi tarah Guna ne sirf do jaanch kar jaana ki 14560 paanchon se divisible hai. 572, 2352, 5600, 6000 aur 77622160 mein se sirf 5600, 6000 aur 77622160 dono test paas karte hain.

Leap years

Koi saal leap year hai agar wo 4 ka multiple ho, sivaay un saalon ke jo 100 se divisible hon par 400 se nahi. Isliye 2000 leap year tha par 1900 nahi. 2024 se 2099 tak 19 leap years hain: 2024, 2028, …, 2096.

4 se divisible palindromes

4 se divisible 4 ank ka palindrome even hona chahiye, isliye uska pehla ank (jo aakhri ke barabar hai) even hai, aur aakhri do ankon se 4 ka multiple banna chahiye. Sabse bada 8888 hai aur sabse chhota 2112.

Prime factorisation se divisibility jaanchne 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.