Modulo Calculator
Type a number and what to divide it by to get a mod n, the quotient and the a = q × n + r check, with negative numbers worked out every common way. Switch to Advanced for power mod, inverses, congruences, a clock picture and a table.
Whole numbers, decimals (use a dot) and negatives all work, up to 120 digits.
How your mod is worked out
Your numbers, step by step: divide, round down, multiply back, take away.
- 1
Divide
17 ÷ 5how many 5s fit?= 3.4
- 2
Round down
⌊3.4⌋drop the decimalsq = 3
- 3
Multiply back
3 × 5the part that fits= 15
- 4
Take away
17 − 15what is left over17 mod 5 = 2
Read more: the formula, a worked example, negative numbers, programming languages, powers and inverses
The modulo (or mod) of two numbers is the remainder after dividing one by the other. You write it a mod n, where n is called the modulus. It is the maths behind clocks (15:00 is 3 o’clock because 15 mod 12 = 3), days of the week, odd and even numbers, check digits and computer code.
The formula
a mod n = a − n × q, where q = a ÷ n rounded down to a whole number
so a = q × n + r, with r between 0 and n (not reaching n)
truncated rule (JavaScript, C, Java %): q = a ÷ n with the decimals cut off
Euclidean rule: r is always 0 or more
power mod: square and multiply through b in binary, taking mod n each time
inverse: x with a × x mod n = 1, found by the extended Euclidean algorithm
Worked example
17 mod 5. Divide: 17 ÷ 5 = 3.4. Round down: q = 3. Multiply back: 3 × 5 = 15. Take away: 17 − 15 = 2. So 17 mod 5 = 2 and the check is 3 × 5 + 2 = 17. On a clock with 5 places (0 to 4), 17 steps from 0 go round 3 full times and land on 2.
With a negative number, −17 mod 5: −17 ÷ 5 = −3.4, and rounding down gives −4 (down means toward minus infinity, not toward 0). Then −4 × 5 = −20 and −17 − (−20) = 3. Check: −4 × 5 + 3 = −17. JavaScript, C and Java cut −3.4 to −3 instead, so −17 % 5 gives −17 − (−15) = −2 there.
Power mod, 1713 mod 5: first 17 mod 5 = 2, and 13 in binary is 1101. Start with 2. Next bit 1: square, 2² = 4, then multiply, 4 × 2 = 8, and 8 mod 5 = 3. Next bit 0: square only, 3² = 9 and 9 mod 5 = 4. Last bit 1: 4² = 16 and 16 mod 5 = 1, then 1 × 2 = 2. So 1713 mod 5 = 2 (1713 is 9,904,578,032,905,937, and it ends in 7, so it leaves 2 after dividing by 5).
Inverse of 17 mod 5: 17 mod 5 = 2. Euclid: 5 = 2 × 2 + 1, then 2 = 2 × 1 + 0, so the gcd is 1 and an inverse exists. Working back: 1 = 5 − 2 × 2, which with the original number is 1 = 7 × 5 − 2 × 17. So 17 × (−2) leaves 1, and −2 mod 5 = 3. Check: 17 × 3 = 51 and 51 mod 5 = 1.
The three rules for negative numbers
| Sum | Floored (maths, Python) | Truncated (JavaScript, C, Java) | Euclidean |
|---|---|---|---|
| 17 mod 5 | 2 | 2 | 2 |
| −7 mod 3 | 2 | −1 | 2 |
| 7 mod −3 | −2 | 1 | 1 |
| −7 mod −3 | −1 | −1 | 2 |
All of them pass the check a = q × n + r; only the quotient q differs. Floored takes the sign of n, truncated takes the sign of a, and Euclidean is never negative. In Python, -7 % 3 is 2; in JavaScript, C and Java, -7 % 3 is −1. To get the maths answer in those languages, use ((a % n) + n) % n.
Where mod is used
- Even or odd: a number is even when n mod 2 = 0.
- Clocks and calendars: 9 o’clock + 5 hours = 14 mod 12 = 2 o’clock; 10 days after a Monday is a Thursday because 10 mod 7 = 3.
- Check digits: ISBN-10 uses mod 11, IBAN bank numbers use mod 97.
- Codes and encryption: RSA uses power mod and inverses mod with numbers hundreds of digits long.
All answers are exact: the calculator works with whole numbers of any size up to 120 digits and turns decimals into whole numbers first (7.5 mod 2 is worked out as 75 mod 20 = 15, which is 1.5), so nothing is rounded.
Common questions
What does mod mean in maths?
a mod n is the remainder when you divide a by n. 17 mod 5 = 2, because 5 goes into 17 three times (15) and 2 is left over. You can check it: 3 × 5 + 2 = 17.
How do you work out a mod n?
Divide a by n, round the answer down to a whole number, multiply that back by n and take it away from a. For 17 mod 5: 17 ÷ 5 = 3.4, round down to 3, 3 × 5 = 15, and 17 − 15 = 2.
What is −7 mod 3?
In maths (and in Python) −7 mod 3 = 2, because −7 = −3 × 3 + 2 and the remainder takes the sign of the number you divide by. In JavaScript, C and Java, −7 % 3 gives −1, because those languages cut the quotient toward zero: −7 = −2 × 3 − 1. Both are correct; they follow different rules.
Is mod the same as remainder?
For positive numbers, yes. They only differ when a number is negative. The maths mod gives an answer with the same sign as n (or 0), while the % remainder in JavaScript, C and Java gives an answer with the same sign as a.
Can you do mod with decimals?
Yes. 7.5 mod 2 = 1.5, because 3 × 2 = 6 and 7.5 − 6 = 1.5. The quotient is still a whole number. This calculator works with exact decimals, so 0.3 mod 0.1 is exactly 0 (a computer’s floating-point maths often says 0.0999…).
What is a modular inverse?
The inverse of a mod n is the number x with a × x mod n = 1. The inverse of 3 mod 11 is 4, because 3 × 4 = 12 = 11 + 1. It only exists when a and n share no factor other than 1, and you find it with the extended Euclidean algorithm.
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