Divisibility Test Calculator

About the Divisibility Test Calculator

The **Divisibility Test Calculator** checks whether a number is divisible by common divisors 2 through 12 — plus any custom divisor you choose — and shows the specific rule that applies in each case. Instead of performing long division, you can work from the same divisibility shortcuts taught in arithmetic and early number theory.

Each divisor has its own rule. Divisibility by 2 checks the last digit; by 3, the digit sum; by 4, the last two digits; by 5, the last digit; by 6, both rules for 2 and 3; by 7, a doubling-and-subtraction method; by 8, the last three digits; by 9, the digit sum; by 10, the final zero; by 11, the alternating digit sum; and by 12, the rules for both 3 and 4. This calculator applies each rule automatically and shows the intermediate values so you can learn and verify the rules yourself.

The **digit sum method** is highlighted prominently — it is the foundation of divisibility tests for 3 and 9, and understanding it deepens your grasp of modular arithmetic. The calculator also shows the exact remainder for each divisor, so even when a number is not divisible, you know how far off it is.

A comprehensive divisibility rules reference table documents every rule from 2 to 12 with examples. The visual divisibility grid gives you a color-coded overview showing which divisors work for the given number. Preset buttons load interesting numbers like 2520 (the smallest number divisible by 1–10), 360, 7919 (a prime), and others. This setup is useful for students learning the rules, teachers preparing examples, and anyone who wants the remainder and rule explanation side by side.

When This Page Helps

Use this page when you want the divisibility rule and the actual remainder together. It is especially helpful for checking classroom examples, puzzle-style number claims, and modular-arithmetic intuition without having to repeat long division for every divisor.

How to Use the Inputs

1. Enter any positive integer to test.
2. Optionally enter a custom divisor beyond 2–12.
3. Click a preset to load an interesting number.
4. Check the divisibility grid for a color-coded overview.
5. Review output cards for digit sum, alternating sum, and custom test.
6. Browse the comprehensive rules table for all divisibility rules.
7. Examine the detailed results table for each divisor's rule application.

Example Calculation

Tips & Best Practices

• Check that all inputs use the same scale and assumptions before trusting the result.
• Compare the answer with the worked example or a rough estimate to catch entry mistakes.

How Divisibility Rules Work

Divisibility rules are shortcuts that let you determine whether one integer divides another without performing full long division. They exploit properties of modular arithmetic. For instance, because 10 ≡ 1 (mod 9), every power of 10 also ≡ 1, so a number and its digit sum leave the same remainder when divided by 9 (or by 3). Similarly, 10 ≡ −1 (mod 11), which means alternating powers of 10 have alternating signs — giving rise to the alternating-digit-sum rule for 11. The rule for 4 checks only the last two digits because 100 ≡ 0 (mod 4), so higher digits contribute nothing to the remainder.

Beyond 2–12: Constructing Custom Rules

For any divisor d, you can build a rule by examining 10^k mod d for successive k values. If the sequence is simple (e.g., constant or alternating), you get a digit-based shortcut. If not, the generic approach is to form telescoping sums or to use the "osculator" method: multiply the last digit by a fixed constant and add to or subtract from the truncated number. While these rules exist for every d, they become unwieldy for larger d; at that point, direct modular arithmetic (long division) is more practical. This calculator lets you test *any* divisor, custom or standard.

Applications of Divisibility in Number Theory

Divisibility is the gateway to deeper concepts: primes (numbers with exactly two divisors), GCD and LCM (built from shared and combined divisors), modular arithmetic (the foundation of cryptography), and Diophantine equations (integer-only solutions). Checking divisibility by small primes is the first step in trial-division factoring. Euler's totient function counts integers less than n that are coprime to n — itself based on divisibility relationships. Practically, divisibility checks appear in check-digit algorithms (ISBN, UPC, credit-card numbers via the Luhn algorithm), hash functions, and load-balancing schemes where tasks are assigned modulo the number of servers.

Frequently Asked Questions

• How does divisibility by 3 work?

  Add up all the digits. If the sum is divisible by 3, the original number is too. For 123: 1+2+3 = 6, and 6 ÷ 3 = 2, so 123 is divisible by 3.

• What is the alternating sum rule for 11?

  Alternate adding and subtracting digits from right to left. If the result is divisible by 11, so is the number. For 2728: 8−2+7−2 = 11, divisible by 11 ✓.

• What is the rule for 7?

  Double the last digit and subtract it from the rest of the number. Repeat until you get a small number. Divisible by 7 at the end → original is too.

• What is the smallest number divisible by 1–10?

  2520. It equals LCM(1,2,3,4,5,6,7,8,9,10). Try it with the 2520 preset!

• Why do digit sum rules work?

  Because 10 ≡ 1 (mod 3) and 10 ≡ 1 (mod 9), so any power of 10 contributes the same as its coefficient. The digit sum preserves the remainder mod 3 and mod 9.

• Can I test divisibility by any number?

  Yes! Enter a custom divisor and the calculator will compute n mod d and show whether the remainder is zero.

• What is the difference between divisibility and factoring?

  Divisibility tests whether a specific d divides n. Factoring finds all prime factors of n. Divisibility is a quick yes/no check; factoring is more comprehensive.