Divisibility Calculator

Number Theory & Classical Divisibility Rules

An integer a is divisible by a non-zero integer b (written b a) if there exists an integer q such that a = b · q, leaving a remainder r = 0:

• **

• Euclidean Division Theorem**:

a = b · q + r (0 r < |b|)

• **

• The Universal Divisibility Rules**:

• Divisible by 2: Last digit is even (0, 2, 4, 6, 8).

• Divisible by 3: Sum of all digits is a multiple of

3.

• Divisible by 4: Last two digits form a number divisible by

4.

• Divisible by 5: Last digit is 0 or

5.

• Divisible by 6: Number satisfies rules for both 2 and

3.

• Divisible by 7: Truncate last digit, double it, and subtract from remaining number; result must be divisible by

7.

• Divisible by 8: Last three digits form a number divisible by

8.

• Divisible by 9: Sum of all digits is divisible by

9.

• Divisible by 10: Last digit is

0.

• Divisible by 11: Alternating sum of digits is divisible by

11.

• Divisible by 12: Number is divisible by both 3 and

4.

How to Use This Calculator

Enter your Dividend Number (A) (e.g. 1260), Divisor Number (B) (e.g. 12), and Arithmetic Case Benchmark — choose from Custom Division Pair, 1,260 ÷ 12 (Exact Quotient: 105 / Remainder 0), 1,024 ÷ 8 (Exact Quotient: 128 / Remainder 0). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Divisibility Calculator

Input variables:

dividendNumber — Dividend Number (A)

divisorNumber — Divisor Number (B)

Computed outputs:

isDivisibleStatus — Divisibility Outcome

integerQuotientValue — Integer Quotient (q)

euclideanRemainderValue — Remainder (r = A mod B)

euclideanIdentityEquation — Euclidean Division Identity

commonDivisorsMatrix — Divisibility by 2 through 13 Check

Mathematical relationships extracted from calculation logic:

q = Math.floor(a / b)

r = ((a % b) + Math.abs(b)) % Math.abs(b)

Standard: NIST / ISO mathematical definitions

Worked Example

Testing whether 1,001 is divisible by 7 yields an exact quotient of 143 with remainder 0, proving that 1,001 = 7 × 11 ×

13.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.