Remainder Calculator (Modulo)
Calculate the remainder when dividing two numbers (modulo operation). See also Long Division Calculator and GCF Calculator.
How to Calculate Remainder
The remainder is what is left over after dividing one number by another. Divide the dividend by the divisor to get the integer quotient, then multiply the quotient by the divisor and subtract from the dividend. The modulo operation (mod) returns this remainder. It is widely used in programming, cryptography, and number theory.
Remainder Formula
A mod B = R
where: A = B × Q + R
Q = floor(A / B) (integer quotient)
R = A − B × Q (remainder)
0 ≤ R < |B|
Example
Calculate: 17 mod 5
Quotient: floor(17 / 5) = floor(3.4) = 3
Remainder: 17 − 5 × 3 = 17 − 15 = 2
Verification: 17 = 5 × 3 + 2 ✓
Frequently Asked Questions
What is the modulo operation?
The modulo operation finds the remainder after division. In programming, it is often written as A % B. For example, 17 % 5 = 2 because 17 divided by 5 is 3 with a remainder of 2.
Can the remainder be negative?
In mathematics, the remainder is typically non-negative. However, some programming languages define modulo differently for negative numbers. This calculator uses the mathematical definition where the remainder is always non-negative.
What is the remainder when dividing by 1?
Any integer divided by 1 has a remainder of 0, since every integer is divisible by 1. For non-integers, the remainder equals the fractional part.
How is modulo used in real life?
Modulo is used in clock arithmetic (12-hour time), determining even/odd numbers, hash functions, cryptography, and cyclic patterns like days of the week.
Solved Examples — Remainder (Modulo)
Example: What day of the week is it 100 days from Monday?
Solution:
Step 1: Days in a week = 7. Calculate 100 mod 7
Step 2: 100 ÷ 7 = 14 remainder 2
Step 3: 2 days after Monday = Wednesday
Answer: Wednesday (100 mod 7 = 2)
Example: You have 247 candies to distribute equally among 12 children. How many are left over?
Solution:
Step 1: 247 mod 12
Step 2: 247 ÷ 12 = 20 remainder 7 (since 12 × 20 = 240)
Step 3: 247 − 240 = 7
Answer: 7 candies left over (each child gets 20)
Example: Is 437 divisible by 9?
Solution:
Step 1: Calculate 437 mod 9
Step 2: 437 ÷ 9 = 48 remainder 5 (since 9 × 48 = 432)
Step 3: Remainder = 437 − 432 = 5 ≠ 0
Answer: No, 437 is not divisible by 9 (remainder = 5)
Practice Questions
Try these on your own:
- Calculate 256 mod 7 (Answer: 4)
- What is 1000 mod 13? (Answer: 12)
- Is 372 even or odd? Use mod 2. (Answer: Even, since 372 mod 2 = 0)
- A clock shows 10:00. What time will it show in 50 hours? (Answer: 12:00, since (10+50) mod 12 = 0 = 12)
- Find the remainder when 2¹⁰ is divided by 7 (Answer: 2, since 1024 mod 7 = 2)
- You need 155 tiles for a floor. Tiles come in boxes of 12. How many are left over? (Answer: 11)
Common Mistakes to Avoid
The most common mistake is confusing the quotient with the remainder. In 17 ÷ 5, the quotient is 3 and the remainder is 2 — students sometimes report 3.4 or just 3. Remember the remainder is ALWAYS less than the divisor. Another error involves negative numbers: different programming languages handle negative modulo differently (Python gives non-negative, C gives sign of dividend), so be careful when coding. A key property to remember is that if the remainder is 0, the number is evenly divisible. Also, A mod B = A when A < B (for positive numbers). Finally, don't confuse the mod operator (%) with the percentage sign (%) — they look the same but mean completely different things.
Key Takeaways
- Remainder = Dividend − (Divisor × Quotient), where Quotient = floor(Dividend/Divisor).
- The remainder is always between 0 and |Divisor| − 1 (inclusive).
- If remainder = 0, the dividend is evenly divisible by the divisor.
- Modulo arithmetic is cyclic — it "wraps around" (like a clock).
- Used in: checking divisibility, clock arithmetic, hash functions, cryptography, and determining even/odd.
- Properties: (a + b) mod n = ((a mod n) + (b mod n)) mod n — useful for large number problems.