"विभाज्यता के नियम": अवतरणों में अंतर

विभाज्यता के नियम उन विधियों को कहते हैं जो सरलता से बता देते हैं कि कोई संख्या किसी दूसरी संख्या
नया पृष्ठ: '''विभाज्यता के नियम''' (divisibility rule) उन विधियों को कहते हैं जो सरलता से बत...
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07:15, 27 अक्टूबर 2009 का अवतरण

विभाज्यता के नियम (divisibility rule) उन विधियों को कहते हैं जो सरलता से बता देते हैं कि कोई संख्या किसी दूसरी संख्या से विभाजित हो सकती है या नहीं। किसी भी आधार वाले संख्या-पद्धति (जैसे, द्वयाधारी या अष्टाधारी संख्याओं) के लिये ऐसे नियम बनाये जा सकते हैं किन्तु यहाँ केवल दासमिक प्रणाली (decimal system) के संख्याओं के लिये नियम दिये गये हैं।

Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also.
विभाजक विभाजन की शर्त/शर्तें उदाहरण
स्वत: सभी पूर्णांक १ से विभाज्य हैं।
संख्या का अन्तिम अंक सम (0, 2, 4, 6, or 8) हो. 1,294: इसमें अन्तिम अंक 4 सम है।
3 दी हुई संख्या के सभी अंकों का योग 3 से विभाजित हो. बहुत बड़ी संख्याओं (जिनके अंकों का योग भी बड़ी संख्या हो) के लिये यह नियम अंकों के योग पर भी लागू किया जाता है। 405:6+3+6=15 जो कि 3 से विभाज्य है. 16,499,205,854,376 के अंकों का योग 69 है; 6 + 9 = 15, 1 + 5 = 6, जो स्पष्टत: 3 से विभाज्य है.
4 Add the ones digit to twice the tens digit. (All digits to the left of the tens digit can be ignored.) 5,096: 6 + (2 × 9) = 24
The last two digits divisible by 4. 40832: 32 is divisible by 4.
If the tens digit is even, and the ones digit is 0, 4, or 8.

If the tens digit is odd, and the ones digit is 2, or 6.

40832: 3 is odd, and the last digit is 2.
5 The last digit is 0 or 5. 490: the last digit is 0.
6 It is divisible by 2 and by 3. 1,458: 1 + 4 + 5 + 8 = 18, 1 + 8 = 9, so it is divisible by 3 and the last digit is even, hence number is divisible 6.
Add the last digit to four times the sum of all other digits. 198: (1 + 9) × 4 + 8 = 48
7 The number obtained from these examples must be divisible by 7, as follows:
Form the alternating sum of blocks of three from right to left. 1,369,851: 851 - 369 + 1 = 483 = 7 × 69
Subtract 2 times the last digit from the rest. 483: 48 - (3 × 2) = 42 = 7 x 6.
Or, add 5 times the last digit to the rest. 483: 48 + (3 × 5) = 63 = 7 x 9.
8 The number obtained from these examples must be divisible by 8, as follows:
If the hundreds digit is even, examine the number formed by the last two digits. 624: 24.
If the hundreds digit is odd, examine the number obtained by the last two digits plus 4. 352: 52 + 4 = 56.
Add the last digit to twice the rest. 56: (5 × 2) + 6 = 16.
Examine the last three digits 34152: Examine divisibility of just 152: 19 x 8
9 The sum of the digits is divisible by 9. For larger numbers, digits may be summed iteratively. The final result must be 9. 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.
10 The last digit is 0. 130: the last digit is 0.
11 The number obtained from these examples must be divisible by 11, as follows:
Form the alternating sum of the digits. 918,082: 9 - 1 + 8 - 0 + 8 - 2 = 22.
Add the digits in blocks of two from right to left. 627: 6 + 27 = 33.
Subtract the last digit from the rest. 627: 62 - 7 = 55.
12 It is divisible by 3 and by 4. 324: it is divisible by 3 and by 4.
Subtract the last digit from twice the rest. 324: (32 × 2) − 4 = 60.
13 The number obtained from these examples must be divisible by 13, as follows:
Add the digits in alternate blocks of three from right to left, then subtract the two sums. 2,911,272: − (2 + 272) + 911 = 637
Add 4 times the last digit to the rest. 637: 63 + (7 × 4) = 91, 9 + (1 × 4) = 13.
14 It is divisible by 2 and by 7. 224: it is divisible by 2 and by 7.
Add the last two digits to twice the rest. The answer must be divisible by 14. 364: (3 × 2) + 64 = 70.
15 It is divisible by 3 and by 5. 390: it is divisible by 3 and by 5.
16 The number obtained from these examples must be divisible by 16, as follows:
If the thousands digit is even, examine the number formed by the last three digits. 254,176: 176.
If the thousands digit is odd, examine the number formed by the last three digits plus 8. 3,408: 408 + 8 = 416.
Sum the number with the last two digits removed, times 4, plus the last two digits. 176: (1 × 4) + 76 = 80.
17 Subtract 5 times the last digit from the rest. 221: 22 - (1 × 5) = 17.
18 It is divisible by 2 and by 9. 342: it is divisible by 2 and by 9.
19 Add twice the last digit to the rest. 437: 43 + (7 × 2) = 57.
20 It is divisible by 10, and the tens digit is even. 360: is divisible by 10, and 6 is even.
If the number formed by the last two digits is divisible by 20. 480: 80 is divisible by 20.

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