visorDivisibility conditionExamples1No special condition. Any integer is divisible by 1.2 is divisible by 1.2The last digit is even (0, 2, 4, 6, or 8).[1][2]1,294: 4 is even.3Sum the digits. If the result is divisible by…
visorDivisibility conditionExamples
1No special condition. Any integer is divisible by 1.2 is divisible by 1.
2The last digit is even (0, 2, 4, 6, or 8).
[1][2]1,294: 4 is even.
3Sum the digits. If the result is divisible by 3, then the original number is divisible by 3.
[1][3][4]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.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.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 3.4Examine the last two digits.[1][2]40832: 32 is divisible by 4.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 6.40832: 3 is odd, and the last digit is 2.Twice the tens digit, plus the ones digit.40832: 2 × 3 + 2 = 8, which is divisible by 4.5The last digit is 0 or 5.[1][2]495: the last digit is 5.6It is divisible by 2 and by 3.[5]1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.7Form the alternating sum of blocks of three from right to left.[4][6]1,369,851: 851 − 369 + 1 = 483 = 7 × 69Subtract 2 times the last digit from the rest. (Works because 21 is divisible by 7.)483: 48 − (3 × 2) = 42 = 7 × 6.Or, add 5 times the last digit to the rest. (Works because 49 is divisible by 7.)483: 48 + (3 × 5) = 63 = 7 × 9.Or, add 3 times the first digit to the next. (This works because 10a + b − 7a = 3a + b − last number has the same remainder)483: 4×3 + 8 = 20 remainder 6, 6×3 + 3 = 21.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.483595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7.8If 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.[1][2]34152: Examine divisibility of just 152: 19 × 8Add four times the hundreds digit to twice the tens digit to the ones digit.34152: 4 × 1 + 5 × 2 + 2 = 169Sum the digits. If the result is divisible by 9, then the original number is divisible by 9.[1][3][4]2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.10The last digit is 0.[2]130: the last digit is 0.11Form the alternating sum of the digits.[1][4]918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22.Add the digits in blocks of two from right to left.[1]627: 6 + 27 = 33.Subtract the last digit from the rest.627: 62 − 7 = 55.If the number of digits is even, add the first and subtract the last digit from the rest.918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11If the number of digits is odd, subtract the first and last digit from the rest.14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 1112It is divisible by 3 and by 4.[5]324: it is divisible by 3 and by 4.Subtract the last digit from twice the rest.324: 32 × 2 − 4 = 60.13Form the alternating sum of blocks of three from right to left.[6]2,911,272: −2 + 911 − 272 = 637Add 4 times the last digit to the rest.637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13.Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): -3, -4, -1, 3, 4, 1 (repeating for digits beyond the hundred-thousands place). Then sum the results.[7]30,747,912: (2 × (-3)) + (1 × (-4)) + (9 × (-1)) + (7 × 3) + (4 × 4) + (7 × 1) + (0 × (-3)) + (3 × (-4)) = 13.14It is divisible by 2 and by 7.[5]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.
1764: 17 × 2 + 64 = 98.15It is divisible by 3 and by 5.[5]390: it is divisible by 3 and by 5.16If 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.Add the last two digits to four times the rest.176: 1 × 4 + 76 = 80.
1168: 11 × 4 + 68 = 112.
Examine the last four digits.[1][2]157,648: 7,648 = 478 × 16.17Subtract 5 times the last digit from the rest.221: 22 − 1 × 5 = 17.18It is divisible by 2 and by 9.[5]342: it is divisible by 2 and by 9.19Add twice the last digit to the rest.437: 43 + 7 × 2 = 57.20It 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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