Tests for Divisibility of Numbers
Let us see whether we can find a pattern that can tell us whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11.
Divisibility by 2
A number Is divisible by 2 if its ones digit Is 0, 2, 4, 6 or 8.
Example 1
Each of the numbers 30. 52, 84, 136, 2108 Is divisible by 2.
Example 2
None of the numbers 71, 83, 215, 467, 629 is divisible by 2.
Divisibility by 3
A number is divisible by 3 If the sum of Its digits is divisible by 3.
Example 1
Consider the number 64275.
Sum of its digits = (6 + 4 + 2 + 7 + 5) = 24, which Is divisible by 3.
Therefore. 64275 is divisible by 3.
Example 1
Consider the number 39583.
Sum of its digits = (3 + 9 + 5 + 8+ 3) = 28, which is not divisible by 3.
Therefore, 39583 is not divisible by 3.
Divisibility by 4
A number is divisible by 4 if the number formed by Its digits in the tens and ones places is divisible by 4.
Example 1
Consider the number 96852.
The number formed by the tens and ones digits is 52, which is divisible by 4.
Therefore, 96852 is divisible by 4.
Example 1
Consider the number 61394.
The number formed by the tens and ones digits is 94, which is not divisible by 4.
Therefore, 61394 is not divisible by 4.
Divisibility by 5
A number Is divisible by 5 if its ones digit is 0 or 5.
Example 1
Each of the numbers 65. 195, 230, 310 is divisible by 5.
Example 1
None of the numbers 71, 83, 94, 106, 327, 148, 279 is divisible by 5.
Divisibility by 6
A number is divisible by 6 if it is divisible by each one of 2 and 3.
NOTE
2 and 3 are the prime factors of 6.
Example 1
Each of the numbers 18, 42, 60, 1 14, 1356 is divisible by 6.
Example 1
None of the numbers 21, 25, 34, 52 is divisible by 6.
Divisibility by 7
A number is divisible by 7 if the difference between twice the ones digit and the number formed by the other digits is either 0 or a multiple of 7.
Example 1
Consider the number 6804.
Clearly, (680 − 2 × 4) = 672, which is divisible by 7.
Therefore, 6804 is divisible by 7.Factors and Multiples
Example 1
Consider the number 137.
Clearly, (2 × 7) − 13 = 1. which is not divisible by 7.
Therefore. 137 is not divisible by 7.
Example 1
Consider die number 1367.
Clearly. 136 − (2 × 7) = 136 − 14 = 122. which is not divisible by 7.
Therefore, 1367 Is not divisible by 7.
Divisibility by 8
A number Is divisible by 8 if the number formed by its digits the hundreds, tens and ones places Is divisible by 8.
Example 1
Consider Hie number 79152.
The number formed by hundreds, tens and ones digits Is 152, which is clearly divisible by 8.
Therefore, 79152 Is divisible by 8.
Example 1
Consider the number 57348.
The number formed by hundreds, tens and ones digits is 348, which is not divisible by 8
Therefore. 57348 is not divisible by 8.
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example 1
Consider the number 65403.
Sum of its digits = (6 + 5 + 4+ 0 + 3) = 18, which is divisible by 9.
Therefore. 65403 is divisible by 9.
Example 1
Consider the number 81326.
Sum of its digits = (8 + 1 + 3 + 2 + 6) = 20, which is not divisible by 9.
Therefore, 81326 is not divisible by 9.
Divisibility by 10
A number is divisible by 10 if its ones digit is 0.
Example 1
Each of the numbers 30. 160. 690, 720 is divisible by 10.
Example 1
None of the numbers 21. 32, 63. 84, etc., is divisible by 10.
Divisibility by 11
A number Is divisible by 11 the difference of the sum of its digits In odd places and the sum of Its digits in even places (starting from the ones place) Is either 0 or a multiple of 11.
| Number | Sum of the digits (at odd places) from the right | Sum of the digits (at even place) from the right | Difference |
|---|---|---|---|
| 90728 | 8 + 7 + 9 = 11 | 2 + 0 = 2 | 24 − 2 = 22 |
| 863423 | 3 + 4 + 6 = 13 | 2 + 3 + 8 = 13 | 13 − 13 = 0 |
| 76844 | 4 + 8 + 7 = 19 | 4 + 6 = 10 | 19 − 10 = 9 |
Example 1
Consider the number 90728.
- Sum of its digits in odd places = (8 + 7 + 9) = 24.
- Sum of its digits in even places = (2 + 0) = 2.
Difference of the two sums = (24 − 2) = 22, which is clearly divisible by 11.
Therefore. 90728 is divisible by 11.
Example 2
Consider the number 863423.
- Sum of its digits in odd places = (3 + 4 + 6) = 13.
- Sum of its digits in even places = (2 + 3 + 8) = 13.
Difference of these sums = (13 − 13) = 0.
Therefore, 863423 is divisible by 1 1 .
Example 3
Consider the number 76844.
- Sum of its digits in odd places = (4 + 8 + 7) = 19.
- Sum of its digits in even places = (4 + 6) = 10.
Difference of these sums = (19 − 10) = 9. which is not divisible by 11.
Therefore, 76844 is not divisible by 11.
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