Divisibility Rules Checker
Instantly check whether a number is divisible by 2–13 or a custom divisor, with plain-English explanations of each rule. Ideal for students learning number theory or anyone verifying factors quickly.
About this calculator
Divisibility is determined by the modulo operation: a number n is divisible by d if and only if n mod d = 0, meaning the remainder of n ÷ d equals zero. Each divisor has a shortcut rule — for example, a number is divisible by 3 if the sum of its digits is divisible by 3, and by 11 if the alternating digit sum is divisible by 11. Under the hood the calculator evaluates n % d for each selected divisor d from the standard set {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} or a custom value. Results are returned as a list of 'divisible' or 'not divisible' verdicts. Understanding these rules builds intuition for factoring, simplifying fractions, and working with prime numbers.
How to use
Suppose you want to check whether 360 is divisible by 2, 3, 5, and 9. Enter 360 as the number, select the standard divisor set, and click Calculate. The calculator evaluates: 360 % 2 = 0 ✓, 360 % 3 = 0 ✓, 360 % 5 = 0 ✓, 360 % 9 = 0 ✓. All four pass. To check a single custom value — say 7 — select 'custom', enter 7, and the tool returns 360 % 7 = 3, so 360 is not divisible by 7.
Frequently asked questions
What are the divisibility rules for numbers 2 through 13?
Each number has a pattern-based shortcut. Divisibility by 2 requires an even last digit; by 3, the digit sum must be divisible by 3; by 4, the last two digits form a number divisible by 4; by 5, the number ends in 0 or 5. For 6, both the 2 and 3 rules must hold; for 7, a specific alternating-block subtraction applies; for 8, the last three digits must be divisible by 8; for 9, the digit sum must be divisible by 9. Rules for 10–13 follow similar digit-manipulation patterns that the calculator applies automatically.
How does the modulo operation determine divisibility?
The modulo operation (written n % d) returns the remainder when n is divided by d. If that remainder is exactly 0, the division is exact and n is divisible by d. For example, 48 % 6 = 0, so 48 is divisible by 6, whereas 50 % 6 = 2, so 50 is not. This single arithmetic check is the universal test behind every divisibility rule.
When would I need to check divisibility by a custom divisor?
Custom divisors are useful when simplifying large fractions, factoring numbers outside the standard set, or solving competition math problems. For instance, to reduce 561/17 you need to confirm 561 % 17 = 0. Custom divisor mode lets you test any integer, not just 2–13, making the calculator suitable for advanced number-theory exercises as well as everyday arithmetic.