visorDivisibility conditionExamples

**1**

No special condition. Any integer is divisible by 1.2 is divisible by 1.

**2**

The last digit is even (0, 2, 4, 6, or 8).

[1][2]1,294: 4 is even.

**3**

Sum 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.

**4**Examine 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.

**5**The last digit is 0 or 5.[1][2]495: the last digit is 5.

**6**It 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.

**7**Form 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 10

*a*+

*b*− 7

*a*= 3

*a*+

*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.

**8**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.[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 = 16

**9**Sum 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.

**10**The last digit is 0.[2]130: the last digit is 0.

**11**Form 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 × 11

**12**It 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.

**13**Form 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.

**14**It 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.

**15**It is divisible by 3 and by 5.[5]390: it is divisible by 3 and by 5.

**16**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.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.**17**Subtract 5 times the last digit from the rest.221: 22 − 1 × 5 = 17.**18**It is divisible by 2 and by 9.[5]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.