What does divisible mean?
One whole number is divisible by another when dividing leaves nothing over. 156 is divisible by 12 because 156 = 12 × 13 exactly. Mathematicians write this as 12 | 156, read as “12 divides 156”.
The quotient and remainder rule
The remainder is always zero or positive, and smaller than the absolute value of the divisor. A remainder of zero means the division is exact.
With negative numbers, the calculator keeps the remainder positive, as number theory does: −43 ÷ 5 gives a quotient of −9 and a remainder of 2, because −43 = 5 × (−9) + 2. Some calculators and programming languages give −8 remainder −3 instead. Both describe the same division, and the sign never changes whether the remainder is zero.
Divisibility rules from 2 to 20
| Divisor | Rule | Example |
|---|---|---|
| 2 | The last digit is even. | 1,438 ends in 8. |
| 3 | The digit sum is divisible by 3. | 4,572 → 4 + 5 + 7 + 2 = 18. |
| 4 | The last two digits are divisible by 4. | 7,316 → 16 ÷ 4 = 4. |
| 5 | The last digit is 0 or 5. | 2,845 ends in 5. |
| 6 | The number passes both the 2 and 3 tests. | 4,572 is even and has digit sum 18. |
| 7 | Remove the last digit and subtract twice it from the rest. Repeat. | 2,051 → 205 − 2 = 203 → 20 − 6 = 14. |
| 8 | The last three digits are divisible by 8. | 12,376 → 376 ÷ 8 = 47. |
| 9 | The digit sum is divisible by 9. | 7,452 → 7 + 4 + 5 + 2 = 18. |
| 10 | The last digit is 0. | 6,190 ends in 0. |
| 11 | The alternating digit sum is divisible by 11. | 5,918 → 8 − 1 + 9 − 5 = 11. |
| 12 | The number passes both the 3 and 4 tests. | 2,316 has digit sum 12 and ends in 16. |
| 13 | Remove the last digit and add four times it to the rest. Repeat. | 5,096 → 509 + 24 = 533 → 53 + 12 = 65. |
| 14 | The number passes both the 2 and 7 tests. | 1,526 is even, and 152 − 12 = 140 → 14 − 0 = 14. |
| 15 | The number passes both the 3 and 5 tests. | 2,745 has digit sum 18 and ends in 5. |
| 16 | The last four digits are divisible by 16. | 53,488 → 3,488 ÷ 16 = 218. |
| 17 | Remove the last digit and subtract five times it from the rest. Repeat. | 3,094 → 309 − 20 = 289 → 28 − 45 = −17. |
| 18 | The number passes both the 2 and 9 tests. | 5,742 is even and has digit sum 18. |
| 19 | Remove the last digit and add twice it to the rest. Repeat. | 2,413 → 241 + 6 = 247 → 24 + 14 = 38. |
| 20 | The last two digits are 00, 20, 40, 60 or 80. | 9,360 ends in 60. |
Testing any other divisor
You can build a test for a bigger divisor by splitting it into parts that share no common factor. 24 is 8 × 3, so a number is divisible by 24 when it passes both the 8 test and the 3 test. The calculator does this automatically for any divisor made from 2, 3, 5, 7, 11, 13, 17 and 19, such as 36, 45, 56 or 72.
The parts must share no factor, which is where people slip up. Splitting 24 into 4 × 6 doesn’t work, because 4 and 6 both contain a 2. The number 12 passes both the 4 and 6 tests, yet it isn’t divisible by 24.
For divisors with no handy rule, such as 23 or 49, the calculator simply divides and shows you the remainder.
Where divisibility is useful
Packaging and equal groups
A warehouse has 1,176 bottles and packs 24 to a case. Since 1,176 ÷ 24 = 49 exactly, it fills 49 full cases with none left over.
Schedules and repeating cycles
A machine takes a reading every 15 seconds. An hour is 3,600 seconds, which divides by 15 exactly, so the hour holds exactly 240 readings with no partial interval at the end.
Fractions, factors and number theory
Divisibility helps you simplify fractions, find common factors, test for primes and check whether something can be shared equally. Both parts of 198/252 divide by 18, so the fraction reduces straight to 11/14.
Large-number divisibility puzzles
Is 7,654,321,098 divisible by 9?
Add its digits: 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 + 9 + 8 = 45. Since 45 is divisible by 9, so is the ten-digit number. The calculator confirms a quotient of 850,480,122 with nothing left over.
Which digit makes 43□58 divisible by 9?
The known digits add up to 4 + 3 + 5 + 8 = 20. The next multiple of 9 is 27, so the missing digit must be 7, giving 43,758. Try the same question with 3 instead of 9 in the missing-digit solver: now 1, 4 and 7 all work, because 21, 24 and 27 are all multiples of 3.
Can 2,051 pass the test for 7 without long division?
Drop the last digit and subtract twice it: 205 − 2 × 1 = 203. Again: 20 − 2 × 3 = 14. Because 14 is divisible by 7, so is 2,051.