Test a number against many divisors
Example data table
| Number | Divisor | Divisible | Reason |
|---|---|---|---|
| 3780 | 2 | Yes | Last digit is 0, so the number is even. |
| 3780 | 3 | Yes | Digit sum is 18, and 18 is divisible by 3. |
| 3780 | 4 | Yes | Last two digits are 80, and 80 is divisible by 4. |
| 3780 | 11 | No | The alternating digit difference is not a multiple of 11. |
| 3780 | 12 | Yes | The number passes both the 3 and 4 tests. |
Formula used
The main rule is: n = d × q + r, where n is the number, d is the divisor, q is the integer quotient, and r is the remainder.
A number is divisible by a divisor exactly when: r = 0. In modular form, that means: n mod d = 0.
Common shortcut rules used by the tool:
- 2: last digit is even.
- 3: digit sum is divisible by 3.
- 4: last two digits are divisible by 4.
- 5: last digit is 0 or 5.
- 6: the number passes both 2 and 3.
- 8: last three digits are divisible by 8.
- 9: digit sum is divisible by 9.
- 10: last digit is 0.
- 11: alternating digit difference is divisible by 11.
- 12: the number passes both 3 and 4.
How to use this calculator
- Enter any whole number in the main input field.
- Add divisors in the list field, separated by commas or spaces.
- Optionally enter a divisor range to test many values quickly.
- Choose ascending or descending sort order.
- Enable the filter if you only want successful tests shown.
- Press Run Divisibility Test to generate the report.
- Review divisibility, remainders, quotients, and rule explanations.
- Use the CSV or PDF buttons to export your displayed results.
Frequently asked questions
1. What does a divisibility test show?
It shows whether a whole number divides evenly by another number. This tool also reports the remainder, the integer quotient, and a short rule explanation.
2. Does a negative sign change divisibility?
No. Divisibility depends on the absolute value of the number. For example, -24 is still divisible by 2, 3, 4, 6, 8, and 12.
3. Can I test many divisors at once?
Yes. You can enter a custom list, add a range, or use both together. Duplicate divisors are automatically removed before calculation.
4. Why do 3 and 9 use the digit sum?
In base ten, a number and its digit sum leave the same remainder when divided by 3 or 9. That makes the digit-sum shortcut reliable.
5. Why is the test for 11 different?
The rule for 11 uses alternating digit sums. If their difference is a multiple of 11, the original number is divisible by 11.
6. What happens when the number is 0?
Zero is divisible by every nonzero divisor. The tool will return a remainder of 0 and an integer quotient of 0 for every tested divisor.
7. Are the quotient values exact?
The quotient shown is the integer quotient from whole-number division. When the remainder is 0, that quotient is also the exact division result.
8. Can this tool handle very large integers?
Yes. The divisibility checks use digit-by-digit string methods, so the tool can process very large whole numbers without relying only on standard integer size limits.