Modulo Calculator

About Modulo Calculator

Calculate a mod b and see the quotient, remainder, and step-by-step working. Handles negative numbers correctly using mathematical modulo (always non-negative) and shows how this differs from the % operator in most programming languages.

What Is the Modulo Operation?

Modulo gives the remainder after integer division. The relationship is:

a = q × b + r where 0 ≤ r < b

Here q is the quotient (floor of a/b) and r is the remainder (a mod b).

Worked example: 17 mod 5

1. Divide: 17 ÷ 5 = 3.4
2. Floor quotient: q = 3
3. Remainder: r = 17 - (3 × 5) = 17 - 15 = 2
4. Result: 17 mod 5 = 2

Modulo Quick Reference

a	b	a mod b	Quotient	Working
23	7	2	3	23 - 3×7 = 23 - 21 = 2
100	12	4	8	100 - 8×12 = 100 - 96 = 4
50	8	2	6	50 - 6×8 = 50 - 48 = 2
15	5	0	3	Evenly divisible
7	10	7	0	7 < 10, so remainder is 7
365	7	1	52	365 days = 52 weeks + 1 day

How Does Modulo Handle Negative Numbers?

This is where programming languages disagree. There are two conventions:

Convention	-7 mod 3	Used By
Mathematical (Euclidean)	2	Mathematics, this calculator
Truncated (% operator)	-1	JavaScript, C, C++, Java
Floored	2	Python, Ruby

Why the difference? It depends on how "quotient" is defined. Truncated division rounds toward zero (so -7/3 = -2), giving remainder -1. Floored division rounds down (so -7/3 = -3), giving remainder 2. Mathematical modulo guarantees the result is always between 0 and b-1.

Converting between them: If the % operator gives a negative result, add b to get the mathematical modulo: -1 + 3 = 2.

Clock Arithmetic - The Most Intuitive Example

A 12-hour clock is modulo 12 in action. Hours "wrap around" after 12:

24-Hour Time	mod 12	12-Hour Clock
0 (midnight)	0 → 12	12:00 AM
9	9	9:00 AM
13	1	1:00 PM
18	6	6:00 PM
23	11	11:00 PM
25 (next day)	1	1:00 AM

Days of the week work the same way with mod 7. If today is Wednesday (day 3) and you count forward 100 days: (3 + 100) mod 7 = 103 mod 7 = 5, which is Friday.

Common Uses of Modulo in Programming

Use Case	Pattern	Example
Check even/odd	n % 2	7 % 2 = 1 (odd), 8 % 2 = 0 (even)
Wrap array index	i % length	Index 7 in array of length 5 → position 2
Cycle through items	counter % n	Alternate 3 colours: counter % 3
Extract last digit	n % 10	1234 % 10 = 4
Check divisibility	n % d == 0	Is 45 divisible by 9? 45 % 9 = 0, yes
Hash table sizing	hash % buckets	Map hash to bucket index
Format every nth item	i % n == 0	Add line break every 5 items

Divisibility Rules Using Modulo

A number is divisible by d if and only if n mod d = 0:

Test	Check	Example
Divisible by 2?	n mod 2 = 0	146 mod 2 = 0 → yes
Divisible by 3?	n mod 3 = 0	246 mod 3 = 0 → yes (digit sum = 12)
Divisible by 5?	n mod 5 = 0	345 mod 5 = 0 → yes
Divisible by 7?	n mod 7 = 0	343 mod 7 = 0 → yes (7³)

Modular Arithmetic in Cryptography

Modular arithmetic is fundamental to modern cryptography. RSA encryption relies on modular exponentiation: computing a^b mod n for very large numbers. Diffie-Hellman key exchange, digital signatures, and hash functions all use modular arithmetic extensively. The security comes from the fact that while computing a^b mod n is fast, reversing the operation (the discrete logarithm problem) is computationally infeasible for large numbers.

For full step-by-step division working, the long division calculator shows every bring-down step. For finding common factors and multiples, the GCF and LCM calculator uses prime factorisation.

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