<P> Note: To test divisibility by any number that can be expressed as 2 or 5, in which n is a positive integer, just examine the last n digits . </P> <P> Note: To test divisibility by any number expressed as the product of prime factors p 1 n p 2 m p 3 q (\ displaystyle p_ (1) ^ (n) p_ (2) ^ (m) p_ (3) ^ (q)), we can separately test for divisibility by each prime to its appropriate power . For example, testing divisibility by 24 (24 = 8 * 3 = 2 * 3) is equivalent to testing divisibility by 8 (2) and 3 simultaneously, thus we need only show divisibility by 8 and by 3 to prove divisibility by 24 . </P> <Table> <Tr> <Th> Divisor </Th> <Th> Divisibility condition </Th> <Th> Examples </Th> </Tr> <Tr> <Td> </Td> <Td> No special condition . Any integer is divisible by 1 . </Td> <Td> 2 is divisible by 1 . </Td> </Tr> <Tr> <Td> </Td> <Td> The last digit is even (0, 2, 4, 6, or 8). </Td> <Td> 1294: 4 is even . </Td> </Tr> <Tr> <Td> </Td> <Td> Sum the digits . The result must be divisible by 3 . </Td> <Td> 405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3 . 16,499,205,854,376 → 1 + 6 + 4 + 9 + 9 + 2 + 0 + 5 + 8 + 5 + 4 + 3 + 7 + 6 sums to 69 → 6 + 9 = 15 → 1 + 5 = 6, which is clearly divisible by 3 . </Td> </Tr> <Tr> <Td> Subtract the quantity of the digits 2, 5, and 8 in the number from the quantity of the digits 1, 4, and 7 in the number . The result must be divisible by 3 . </Td> <Td> Using the example above: 16,499,205,854,376 has four of the digits 1, 4 and 7 and four of the digits 2, 5 and 8; ∴ Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3 . </Td> </Tr> <Tr> <Td> </Td> <Td> The last two digits form a number that is divisible by 4 . </Td> <Td> 40,832: 32 is divisible by 4 . </Td> </Tr> <Tr> <Td> If the tens digit is even, the ones digit must be 0, 4, or 8 . If the tens digit is odd, the ones digit must be 2 or 6 . </Td> <Td> 40,832: 3 is odd, and the last digit is 2 . </Td> </Tr> <Tr> <Td> Twice the tens digit, plus the ones digit is divisible by 4 . </Td> <Td> 40832: 2 × 3 + 2 = 8, which is divisible by 4 . </Td> </Tr> <Tr> <Td> 5 </Td> <Td> The last digit is 0 or 5 . </Td> <Td> 495: the last digit is 5 . </Td> </Tr> <Tr> <Td> 6 </Td> <Td> It is divisible by 2 and by 3 . </Td> <Td> 1458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6 . </Td> </Tr> <Tr> <Td> 7 </Td> <Td> Forming an alternating sum of blocks of three from right to left gives a multiple of 7 </Td> <Td> 1,369,851: 851 − 369 + 1 = 483 = 7 × 69 </Td> </Tr> <Tr> <Td> Subtracting 2 times the last digit from the rest gives a multiple of 7 . (Works because 21 is divisible by 7 .) </Td> <Td> 483: 48 − (3 × 2) = 42 = 7 × 6 . </Td> </Tr> <Tr> <Td> Adding 5 times the last digit to the rest gives a multiple of 7 . (Works because 49 is divisible by 7 .) </Td> <Td> 483: 48 + (3 × 5) = 63 = 7 × 9 . </Td> </Tr> <Tr> <Td> Adding 3 times the first digit to the next gives a multiple of 7 (This works because 10a + b − 7a = 3a + b − last number has the same remainder) </Td> <Td> 483: 4 × 3 + 8 =' 20' remainder 6, <P> 203: 2 × 3 + 0 =' 6' </P> <P> 63: 6 × 3 + 3 = 21 . </P> </Td> </Tr> <Tr> <Td> Adding the last two digits to twice the rest gives a multiple of 7 . (Works because 98 is divisible by 7 .) </Td> <Td> 483,595: 95 + (2 × 4835) = 9765: 65 + (2 × 97) = 259: 59 + (2 × 2) = 63 . </Td> </Tr> <Tr> <Td> Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, - 1, - 3, - 2 (repeating for digits beyond the hundred - thousands place). Then sum the results . </Td> <Td> 483,595: (4 × (- 2)) + (8 × (- 3)) + (3 × (- 1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7 . </Td> </Tr> <Tr> <Td> Adding the last digit to 3 times the rest gives a multiple of 7 . </Td> <Td> 224: 4 + (3 x 22) = 70 </Td> </Tr> <Tr> <Td> Adding 3 times the last digit to 2 times the rest gives a multiple of 7 . </Td> <Td> 245: (3 x 5) + (2 x 24) = 7 x 9 = 63 </Td> </Tr> <Tr> <Td> 8 </Td> <Td> If the hundreds digit is even, the number formed by the last two digits must be divisible by 8 . </Td> <Td> 624: 24 . </Td> </Tr> <Tr> <Td> If the hundreds digit is odd, the number obtained by the last two digits plus 4 must be divisible by 8 . </Td> <Td> 352: 52 + 4 = 56 . </Td> </Tr> <Tr> <Td> Add the last digit to twice the rest . The result must be divisible by 8 . </Td> <Td> 56: (5 × 2) + 6 = 16 . </Td> </Tr> <Tr> <Td> The last three digits are divisible by 8 . </Td> <Td> 34,152: Examine divisibility of just 152: 19 × 8 </Td> </Tr> <Tr> <Td> Add four times the hundreds digit to twice the tens digit to the ones digit . The result must be divisible by 8 . </Td> <Td> 34,152: 4 × 1 + 5 × 2 + 2 = 16 </Td> </Tr> <Tr> <Td> 9 </Td> <Td> Sum the digits . The result must be divisible by 9 . </Td> <Td> 2880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9 . </Td> </Tr> <Tr> <Td> 10 </Td> <Td> The ones digit is 0 . </Td> <Td> 130: the ones digit is 0 . </Td> </Tr> <Tr> <Td> 11 </Td> <Td> Form the alternating sum of the digits . The result must be divisible by 11 . </Td> <Td> 918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22 = 2 × 11 . </Td> </Tr> <Tr> <Td> Add the digits in blocks of two from right to left . The result must be divisible by 11 . </Td> <Td> 627: 6 + 27 = 33 = 3 × 11 . </Td> </Tr> <Tr> <Td> Subtract the last digit from the rest . The result must be divisible by 11 . </Td> <Td> 627: 62 − 7 = 55 = 5 × 11 . </Td> </Tr> <Tr> <Td> Add the last digit to the hundreds place (add 10 times the last digit to the rest). The result must be divisible by 11 . </Td> <Td> 627: 62 + 70 = 132: 13 + 20 = 33 = 3 × 11 . </Td> </Tr> <Tr> <Td> If the number of digits is even, add the first and subtract the last digit from the rest . The result must be divisible by 11 . </Td> <Td> 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11 </Td> </Tr> <Tr> <Td> If the number of digits is odd, subtract the first and last digit from the rest . The result must be divisible by 11 . </Td> <Td> 14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11 </Td> </Tr> <Tr> <Td> 12 </Td> <Td> It is divisible by 3 and by 4 . </Td> <Td> 324: it is divisible by 3 and by 4 . </Td> </Tr> <Tr> <Td> Subtract the last digit from twice the rest . The result must be divisible by 12 . </Td> <Td> 324: 32 × 2 − 4 = 60 = 5 × 12 . </Td> </Tr> <Tr> <Td> 13 </Td> <Td> Form the alternating sum of blocks of three from right to left . </Td> <Td> 2,911,272: 2 - 911 + 272 = - 637 </Td> </Tr> <Tr> <Td> Add 4 times the last digit to the rest . The result must be divisible by 13 . </Td> <Td> 637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13 . </Td> </Tr> <Tr> <Td> Subtract the last two digits from four times the rest . The result must be divisible by 13 . </Td> <Td> 923: 9 × 4 - 23 = 13 . </Td> </Tr> <Tr> <Td> Subtract 9 times the last digit from the rest . The result must be divisible by 13 . </Td> <Td> 637: 63 - 7 × 9 = 0 . </Td> </Tr> <Tr> <Td> 14 </Td> <Td> It is divisible by 2 and by 7 . </Td> <Td> 224: it is divisible by 2 and by 7 . </Td> </Tr> <Tr> <Td> Add the last two digits to twice the rest . The result must be divisible by 14 . </Td> <Td> 364: 3 × 2 + 64 = 70 . 1764: 17 × 2 + 64 = 98 . </Td> </Tr> <Tr> <Td> 15 </Td> <Td> It is divisible by 3 and by 5 . </Td> <Td> 390: it is divisible by 3 and by 5 . </Td> </Tr> <Tr> <Td> 16 </Td> <Td> If the thousands digit is even, examine the number formed by the last three digits . </Td> <Td> 254,176: 176 . </Td> </Tr> <Tr> <Td> If the thousands digit is odd, examine the number formed by the last three digits plus 8 . </Td> <Td> 3408: 408 + 8 = 416 . </Td> </Tr> <Tr> <Td> Add the last two digits to four times the rest . The result must be divisible by 16 . </Td> <Td> 176: 1 × 4 + 76 = 80 . <P> 1168: 11 × 4 + 68 = 112 . </P> </Td> </Tr> <Tr> <Td> Examine the last four digits . </Td> <Td> 157,648: 7,648 = 478 × 16 . </Td> </Tr> <Tr> <Td> 17 </Td> <Td> Subtract 5 times the last digit from the rest . </Td> <Td> 221: 22 − 1 × 5 = 17 . </Td> </Tr> <Tr> <Td> Subtract the last two digits from two times the rest . </Td> <Td> 4,675: 46 × 2 - 75 = 17 . </Td> </Tr> <Tr> <Td> 18 </Td> <Td> It is divisible by 2 and by 9 . </Td> <Td> 342: it is divisible by 2 and by 9 . </Td> </Tr> <Tr> <Td> 19 </Td> <Td> Add twice the last digit to the rest . </Td> <Td> 437: 43 + 7 × 2 = 57 . </Td> </Tr> <Tr> <Td> Add 4 times the last two digits to the rest . </Td> <Td> 6935: 69 + 35 × 4 = 209 . </Td> </Tr> <Tr> <Td> 20 </Td> <Td> It is divisible by 10, and the tens digit is even . </Td> <Td> 360: is divisible by 10, and 6 is even . </Td> </Tr> <Tr> <Td> The number formed by the last two digits is divisible by 20 . </Td> <Td> 480: 80 is divisible by 20 . </Td> </Tr> <Tr> <Td> 21 </Td> <Td> Subtract twice the last digit from the rest . </Td> <Td> 168: 16 − 8 × 2 = 0 . </Td> </Tr> <Tr> <Td> It is divisible by 3 and by 7 . </Td> <Td> 231: it is divisible by 3 and by 7 . </Td> </Tr> <Tr> <Td> 22 </Td> <Td> It is divisible by 2 and by 11 . </Td> <Td> 352: it is divisible by 2 and by 11 . </Td> </Tr> <Tr> <Td> 23 </Td> <Td> Add 7 times the last digit to the rest . </Td> <Td> 3128: 312 + 8 × 7 = 368 . 36 + 8 × 7 = 92 . </Td> </Tr> <Tr> <Td> Add 3 times the last two digits to the rest . </Td> <Td> 1725: 17 + 25 × 3 = 92 . </Td> </Tr> <Tr> <Td> 24 </Td> <Td> It is divisible by 3 and by 8 . </Td> <Td> 552: it is divisible by 3 and by 8 . </Td> </Tr> <Tr> <Td> 25 </Td> <Td> The number formed by the last two digits is divisible by 25 . </Td> <Td> 134,250: 50 is divisible by 25 . </Td> </Tr> <Tr> <Td> 26 </Td> <Td> It is divisible by 2 and by 13 . </Td> <Td> 156: it is divisible by 2 and by 13 . </Td> </Tr> <Tr> <Td> 27 </Td> <Td> Sum the digits in blocks of three from right to left . The result must be divisible by 27 . </Td> <Td> 2,644,272: 2 + 644 + 272 = 918 . </Td> </Tr> <Tr> <Td> Subtract 8 times the last digit from the rest . The result must be divisible by 27 . </Td> <Td> 621: 62 − 1 × 8 = 54 . </Td> </Tr> <Tr> <Td> Subtract the last two digits from 8 times the rest . The result must be divisible by 27 . </Td> <Td> 6507: 65 × 8 - 7 = 520 - 7 = 513 = 27 × 19 . </Td> </Tr> <Tr> <Td> 28 </Td> <Td> It is divisible by 4 and by 7 . </Td> <Td> 140: it is divisible by 4 and by 7 . </Td> </Tr> <Tr> <Td> 29 </Td> <Td> Add three times the last digit to the rest . The result must be divisible by 29 . </Td> <Td> 348: 34 + 8 × 3 = 58 . </Td> </Tr> <Tr> <Td> Add 9 times the last two digits to the rest . The result must be divisible by 29 . </Td> <Td> 5510: 55 + 10 × 9 = 145 = 5 × 29 . </Td> </Tr> <Tr> <Td> 30 </Td> <Td> It is divisible by 3 and by 10 . </Td> <Td> 270: it is divisible by 3 and by 10 . </Td> </Tr> </Table> <Tr> <Th> Divisor </Th> <Th> Divisibility condition </Th> <Th> Examples </Th> </Tr>

What is the rule for divisibility by 12
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