Division with Remainder

Calculator for computing quotient and modulo for integer division

Division with Remainder Calculator

Division with Remainder (Modulo operation)

Computes for integer division the quotient and the remainder (modulo) when division is not exact

Basic principle

Dividend ÷ Divisor = Quotient remainder Modulo
Example: 11 ÷ 4 = 2 remainder 3, because 2 × 4 + 3 = 11

Dividend
The number that will be divided
Divisor
The number to divide by
Note: Divisor must not be 0
Division display
11 ÷ 4 = 2 remainder 3
Verification: 2 × 4 + 3 = 8 + 3 = 11 ✓
Division with Remainder result
Quotient:
—
Remainder (Modulo):
—
Enter dividend and divisor and click "Compute"

Division Info

Division with Remainder

Fundamental property: a = b × q + r

Dividend Quotient Modulo

Quotient: How many times b fits into a
Remainder: What is left (0 ≤ r < |b|)

Quick examples
15 ÷ 4: Quotient = 3, Remainder = 3
20 ÷ 5: Quotient = 4, Remainder = 0
17 ÷ 6: Quotient = 2, Remainder = 5
Properties
• Remainder is always < |Divisor|
• Remainder ≥ 0 for positive numbers
• If remainder = 0, then divisible
• Unique solution


Mathematical foundations of division with remainder

Division algorithm
For integers a and b ≠ 0:
\[a = bq + r\]
where 0 ≤ r < |b|

Unique representation with quotient q and remainder r

Modulo operation
\[a \bmod b = r\]
The remainder of the division a ÷ b

Basis for modular arithmetic

Euclidean algorithm
Division with remainder as basis for:
\[\gcd(a,b) = \gcd(b, a \bmod b)\]

Efficient computation of greatest common divisor

Negative numbers
Signs affect the result:
-7 ÷ 3 = -3 remainder 2
7 ÷ -3 = -3 remainder -2

Different conventions possible

Step-by-step examples and methods

Example 1: Simple division
17 ÷ 5
17 ÷ 5 = 3 remainder 2
Verification: 3 × 5 + 2 = 17
Quotient: 3, Remainder: 2

17 is not a multiple of 5

Example 2: Exact division
20 ÷ 4
20 ÷ 4 = 5 remainder 0
Verification: 5 × 4 + 0 = 20
Quotient: 5, Remainder: 0

20 is a multiple of 4

Example 3: Negative numbers
-7 ÷ 3
-7 ÷ 3 = -3 remainder 2
Verification: (-3) × 3 + 2 = -7
Quotient: -3, Remainder: 2

Watch signs

Schriftliche Division: 145 ÷ 3
145 : 3 = 48 remainder 1
145 : 3 = 48
14↓   ← 14 ÷ 3 = 4
12    ← 4 × 3 = 12
---
 25 ← 14 - 12 = 2, bring down 5
 25 ÷ 3 = 8
 24 ← 8 × 3 = 24
---
  1 ← 25 - 24 = 1 (remainder)
Step 1
14 ÷ 3 = 4 remainder 2
Use the first two digits
Step 2
25 ÷ 3 = 8 remainder 1
Bring down the next digit
Result
145 ÷ 3 = 48 remainder 1
Verification: 48×3+1=145

Long division shows the algorithm step by step

Applications of division with remainder

Division with remainder is a fundamental concept with wide applications:

Mathematics & number theory
  • Euclidean algorithm (gcd computation)
  • Modular arithmetic and congruences
  • Divisibility tests
  • Number theory and primality tests
Computer Science & programming
  • Hash functions and data structures
  • Circular buffers and cyclic structures
  • Cryptography (RSA, ECC)
  • Pseudo-random number generation
Time calculations & calendars
  • Weekday and calendar computations
  • Timezone conversion
  • Periodic event calculations
  • Leap years and cycles
Practical applications
  • Distributing objects into groups
  • Rounding and approximation
  • Resource planning and allocation
  • Quality control and sampling

Division with remainder: Foundation of discrete mathematics

Division with remainder is one of the most fundamental concepts in mathematics and forms the core of number theory. This simple operation—splitting a number into equal parts with a possible leftover—reveals deep mathematical structures and enables elegant solutions to complex problems. From the Euclidean algorithm to modular arithmetic and modern cryptography, division with remainder pervades discrete mathematics. The unique representation a = bq + r with 0 ≤ r < |b| is not only a computational tool but also reveals the underlying order of integers.

Summary

Division with remainder exemplifies the elegance of mathematical precision: a simple partitioning evolves into a powerful tool for complex computations and theoretical insights. From practical distribution problems to the algorithmic power of the Euclidean algorithm and cryptographic security, division with remainder demonstrates how basic arithmetic operations support advanced mathematical methods.