Modulo Calculator - Calculate Remainder & Mod Operations Online
Result:
17 mod 5 = 2
Quotient: 3
Formula: 17 = 5 × 3 + 2
How Modulo Operation Works
Input Numbers
Enter dividend and divisor
Divide
Perform integer division
Find Remainder
Calculate what's left over
Modulo Formulas & Calculations
Basic Formula
a mod b = a - b × ⌊a/b⌋
Where ⌊⌋ is floor function
Example: 17 mod 5 = 17 - 5 × ⌊17/5⌋ = 17 - 5 × 3 = 2
Division Algorithm
a = bq + r
Where q is quotient, r is remainder
Example: 17 = 5 × 3 + 2
Negative Numbers
(-a) mod b = b - (a mod b)
For negative dividends
Example: (-17) mod 5 = 5 - (17 mod 5) = 5 - 2 = 3
Properties
(a + b) mod n = ((a mod n) + (b mod n)) mod n
Modular addition property
Example: (15 + 8) mod 7 = ((15 mod 7) + (8 mod 7)) mod 7
Modulo Calculation Table
| Dividend (a) | Divisor (b) | Quotient (q) | Remainder (r) | Verification |
|---|---|---|---|---|
| 10 | 3 | 3 | 1 | 10 = 3 × 3 + 1 |
| 15 | 4 | 3 | 3 | 15 = 4 × 3 + 3 |
| 20 | 6 | 3 | 2 | 20 = 6 × 3 + 2 |
| 25 | 7 | 3 | 4 | 25 = 7 × 3 + 4 |
| 30 | 8 | 3 | 6 | 30 = 8 × 3 + 6 |
| 35 | 9 | 3 | 8 | 35 = 9 × 3 + 8 |
| 40 | 11 | 3 | 7 | 40 = 11 × 3 + 7 |
| 45 | 12 | 3 | 9 | 45 = 12 × 3 + 9 |
| 50 | 13 | 3 | 11 | 50 = 13 × 3 + 11 |
| 55 | 14 | 3 | 13 | 55 = 14 × 3 + 13 |
| 60 | 16 | 3 | 12 | 60 = 16 × 3 + 12 |
| 65 | 17 | 3 | 14 | 65 = 17 × 3 + 14 |
| 70 | 18 | 3 | 16 | 70 = 18 × 3 + 16 |
| 75 | 19 | 3 | 18 | 75 = 19 × 3 + 18 |
| 80 | 21 | 3 | 17 | 80 = 21 × 3 + 17 |
Modulo Operations Progression Chart
10 mod 3
15 mod 4
20 mod 6
25 mod 7
30 mod 8
35 mod 9
Practice Problems
Problem 1:
Calculate 23 mod 7
Solution: 23 ÷ 7 = 3 remainder 2, so 23 mod 7 = 2
Problem 2:
Calculate 100 mod 13
Solution: 100 ÷ 13 = 7 remainder 9, so 100 mod 13 = 9
Problem 3:
Calculate 45 mod 8
Solution: 45 ÷ 8 = 5 remainder 5, so 45 mod 8 = 5
Problem 4:
Calculate 128 mod 15
Solution: 128 ÷ 15 = 8 remainder 8, so 128 mod 15 = 8
Problem 5:
Calculate 200 mod 17
Solution: 200 ÷ 17 = 11 remainder 13, so 200 mod 17 = 13
Daily Uses of Modulo Operations
Clock arithmetic uses mod 12 for hours and mod 60 for minutes
Computer graphics use modulo for wrapping textures and animations
Hash tables use modulo to distribute data across buckets
Cryptography relies on modular arithmetic for secure encryption
Calendar calculations use modulo to determine day of week