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1. Diviso
Divisibility condition Examples
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1 Automatic. Any integer is divisible by 1.
2 The last digit is even (0, 2, 4, 6, or 8).[1][2] 1,294: 4 is even.
405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly
divisible by 3.
Sum the digits.[1][3][4]
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.
3
Using the example above: 16,499,205,854,376 has four of the digits 1,
Subtract the quantity of the digits 2, 5 and 8 in the number
4 and 7; four of the digits 2, 5 and 8; ∴ Since 4 − 4 = 0 is a multiple of
from the quantity of the digits 1, 4 and 7 in the number.
3, the number 16,499,205,854,376 is divisible by 3.
Examine the last two digits.[1][2] 40832: 32 is divisible by 4.
If the tens digit is even, and the ones digit is 0, 4, or 8.
4 40832: 3 is odd, and the last digit is 2.
If the tens digit is odd, and the ones digit is 2 or 6.
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,

2. hence the number is divisible by 6.
Form the alternating sum of blocks of three from right to
1,369,851: 851 − 369 + 1 = 483 = 7 × 69
left.[4][6]
Subtract 2 times the last digit from the rest. (Works
483: 48 − (3 × 2) = 42 = 7 × 6.
because 21 is divisible by 7.)
Or, add 5 times the last digit to the rest. (Works because
483: 48 + (3 × 5) = 63 = 7 × 9.
49 is divisible by 7.)
7
Or, add 3 times the first digit to the next. (This works
because 10a + b − 7a = 3a + b − last number has the 483: 4×3 + 8 = 20 remainder 6, 6×3 + 3 = 21.
same remainder)
Multiply each digit (from right to left) by the digit in the
corresponding position in this pattern (from left to right): 1, 483595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) =
3, 2, -1, -3, -2 (repeating for digits beyond the hundred- 7.
thousands place). Then sum the results.
If the hundreds digit is even, examine the number formed
624: 24.
by the last two digits.
8 If the hundreds digit is odd, examine the number obtained
352: 52 + 4 = 56.
by the last two digits plus 4.
Add the last digit to twice the rest. 56: (5 × 2) + 6 = 16.

3. Examine the last three digits[1][2] 34152: Examine divisibility of just 152: 19 × 8
Add four times the hundreds digit to twice the tens digit to
34152: 4 × 1 + 5 × 2 + 2 = 16
the ones digit.
9 Sum the digits.[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.
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.
11
If the number of digits is even, add the first and subtract 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 +
the last digit from the rest. 1 − 5 = 77 = 7 × 11
If the number of digits is odd, subtract the first and last
14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11
digit from the rest.
It is divisible by 3 and by 4.[5] 324: it is divisible by 3 and by 4.
12
Subtract the last digit from twice the rest. 324: 32 × 2 − 4 = 60.

4. Form the alternating sum of blocks of three from right to
2,911,272: −2 + 911 − 272 = 637
left.[6]
Add 4 times the last digit to the rest. 637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13.
13
Multiply each digit (from right to left) by the digit in the
corresponding position in this pattern (from left to right): -3, 30,747,912: (2 × (-3)) + (1 × (-4)) + (9 × (-1)) + (7 × 3) + (4 × 4) + (7 × 1)
-4, -1, 3, 4, 1 (repeating for digits beyond the hundred- + (0 × (-3)) + (3 × (-4)) = 13.
thousands place). Then sum the results.[7]
It is divisible by 2 and by 7.[5] 224: it is divisible by 2 and by 7.
14
Add the last two digits to twice the rest. The answer must
364: 3 × 2 + 64 = 70.
be divisible by 14.
15 It is divisible by 3 and by 5.[5] 390: it is divisible by 3 and by 5.
If the thousands digit is even, examine the number formed
254,176: 176.
by the last three digits.
If the thousands digit is odd, examine the number formed
3,408: 408 + 8 = 416.
by the last three digits plus 8.
16
176: 1 × 4 + 76 = 80.
Add the last two digits to four times the rest.
1168: 11 × 4 + 68 = 112.
Examine the last four digits.[1][2] 157,648: 7,648 = 478 × 16.

5. 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.
It is divisible by 10, and the tens digit is even. 360: is divisible by 10, and 6 is even.
20 If the number formed by the last two digits is divisible by
480: 80 is divisible by 20.
20.