Congruence Modulo Calculator (a ≡ b mod n)

This calculator checks a congruence a ≡ b (mod n). Enter two whole numbers a and b and the modulus n; you get a mod n, b mod n and (a − b) mod n. The two numbers are congruent when the last result is 0.

Formula

a mod n = a − n × ⌊a ÷ n⌋, the remainder between 0 and n − 1. a ≡ b (mod n) exactly when n divides a − b, that is when (a − b) mod n = 0.

• a, b: whole numbers, positive or negative
• n: the modulus, a whole number of at least 1
• ⌊ ⌋ means round down

Worked Example

Is 47 ≡ 17 (mod 5)?

1. 47 ÷ 5 = 9.4, rounded down 9, so 47 mod 5 = 47 − 45 = 2
2. 17 ÷ 5 = 3.4, rounded down 3, so 17 mod 5 = 17 − 15 = 2
3. (47 − 17) mod 5 = 30 mod 5 = 0
4. The remainders match and the difference mod 5 is 0, so 47 ≡ 17 (mod 5).

How to read the result

Clock arithmetic is mod 12 (or 24), days of the week are mod 7, and check digits use mod 10 or mod 11. This calculator always returns a remainder from 0 to n − 1, which is the mathematician’s convention, so −7 mod 3 = 2.

Limits and common mistakes

• Enter whole numbers. Decimal inputs give a decimal “remainder” that has no meaning in congruences.
• Many programming languages give −1 for −7 % 3; this page gives 2.
• n = 1 makes every pair congruent.

Frequently Asked Questions

What is 100 mod 7?

2, because 7 × 14 = 98.

What does ≡ mean?

Congruent: same remainder after division by n.

What is −7 mod 3?

2, because −7 = 3 × (−3) + 2.

Is 0 a valid remainder?

Yes. a mod n = 0 means n divides a.

in a | a | - | 47 | -
in b | b | - | 17 | -
in n | Modulus n | - | 5 | 1
out r | a mod n | - | a - n * floor(a / n) | 0
out rb | b mod n | - | b - n * floor(b / n) | 0
out dm | (a − b) mod n (0 = congruent) | - | (a - b) - n * floor((a - b) / n) | 0