Long Division with Remainder
Complete guide to division with remainder, step-by-step algorithms, and negative numbers
Introduction to Division with Remainder
Division with remainder (also called integer division or Euclidean division) is the process of dividing two integers to produce a quotient and a remainder. This fundamental operation is essential in number theory, cryptography, and computer science.
When a natural number \(a\) (the dividend) is divided by a natural number \(b\) (the divisor), we calculate how many times \(b\) is contained in \(a\). The result consists of a quotient \(q\) and possibly a remainder \(r\).
- \(\displaystyle a\) = Dividend (the number being divided)
- \(\displaystyle b\) = Divisor (the number you divide by)
- \(\displaystyle q\) = Quotient (the result of the division)
- \(\displaystyle r\) = Remainder (what's left over)
The Division Algorithm
The relationship between dividend, divisor, quotient, and remainder is expressed by the Division Algorithm:
\(\displaystyle a = b \cdot q + r\)
where \(\displaystyle 0 \leq r < |b|\)
The dividend equals the divisor times the quotient plus the remainder. The remainder must be non-negative and less than the absolute value of the divisor.
Basic Example: 17 ÷ 5
Verification using the formula:
\(\displaystyle 17 = 5 \cdot 3 + 2\)
\(\displaystyle 17 = 15 + 2\) ✓
Calculating the Remainder
The remainder is the difference between the dividend and the largest multiple of the divisor that doesn't exceed the dividend:
\(\displaystyle r = a - (b \cdot q)\)
or \(\displaystyle r = a \mod b\) (modulo operation)
Finding the Remainder
For \(\displaystyle 17 \div 5\):
A remainder occurs only when the dividend is not a multiple of the divisor. In other words, when the dividend is not evenly divisible by the divisor. If the remainder is 0, then \(a\) is divisible by \(b\).
Division with Negative Numbers
When dividing numbers with different signs, the quotient and remainder follow specific rules.
| Division | Quot. | Rem. | Verification |
|---|---|---|---|
| \(\displaystyle 7 \div 3\) | \(\displaystyle 2\) | \(\displaystyle 1\) | \(\displaystyle 3 \cdot 2 + 1 = 7\) ✓ |
| \(\displaystyle -7 \div 3\) | \(\displaystyle -2\) | \(\displaystyle -1\) | \(\displaystyle 3 \cdot (-2) + (-1) = -7\) ✓ |
| \(\displaystyle 7 \div (-3)\) | \(\displaystyle -2\) | \(\displaystyle 1\) | \(\displaystyle (-3) \cdot (-2) + 1 = 7\) ✓ |
| \(\displaystyle -7 \div (-3)\) | \(\displaystyle 2\) | \(\displaystyle -1\) | \(\displaystyle (-3) \cdot 2 + (-1) = -7\) ✓ |
Quotient sign: Same as multiplication — positive if both have the same sign,
negative if they have different signs.
Remainder sign: Takes the sign of the dividend.
Long Division: Step-by-Step Algorithm
Long division is a systematic method for dividing multi-digit numbers. Let's work through a complete example: \(\displaystyle 145 \div 3\)
1 Set up the division
Write the dividend and divisor side by side with the division symbol:
2 Divide the first digit(s)
Start from the left. Can 1 be divided by 3? No (1 < 3). So take the first two digits: 14.
How many times does 3 go into 14?
Write 4 as the first digit of the quotient.
3 Multiply and subtract
Multiply the quotient digit by the divisor, write it below, and subtract:
4 Bring down the next digit
Bring down the next digit (5) next to the remainder (2) to form 25:
Bring down \(\displaystyle 5\): forms \(\displaystyle 25\)
5 Repeat the process
How many times does 3 go into 25?
Subtract: \(\displaystyle 25 - 24 = 1\)
6 Final Result
No more digits to bring down. The remainder is 1.
\(\displaystyle 145 \div 3 = 48\) remainder \(\displaystyle 1\)
Verification: \(\displaystyle 3 \times 48 + 1 = 144 + 1 = 145\) ✓
Additional Examples
Example 1: Division with No Remainder
Calculate: 156 ÷ 12
Verification: \(\displaystyle 12 \times 13 = 156\) ✓
Since the remainder is 0, we say 156 is divisible by 12.
Example 2: Larger Division
Calculate: 1234 ÷ 7
Verification: \(\displaystyle 7 \times 176 + 2 = 1232 + 2 = 1234\) ✓
Example 3: Division by a Larger Divisor
Calculate: 45 ÷ 7
Verification: \(\displaystyle 7 \times 6 + 3 = 42 + 3 = 45\) ✓
Properties and Applications
Key Properties
- The remainder \(\displaystyle r\) always satisfies \(\displaystyle 0 \leq r < |b|\)
- If \(\displaystyle r = 0\), then \(\displaystyle a\) is divisible by \(\displaystyle b\)
- The quotient and remainder are unique for given \(\displaystyle a\) and \(\displaystyle b > 0\)
- Division by zero is undefined
Applications
- Computer Science: Hash functions, modular arithmetic
- Cryptography: RSA encryption relies on modular division
- Calendar Calculations: Finding day of the week
- Clock Arithmetic: 12-hour and 24-hour conversions
- Error Detection: Check digits, ISBN validation
- Number Theory: Prime factorization, GCD calculations
The Modulo Operation
The modulo operation (written as \(\displaystyle a \mod b\) or \(\displaystyle a \% b\)) returns the remainder when \(\displaystyle a\) is divided by \(\displaystyle b\).
\(\displaystyle a \mod b = r\) where \(\displaystyle a = b \cdot q + r\)
Examples of Modulo
- \(\displaystyle 17 \mod 5 = 2\)
- \(\displaystyle 20 \mod 4 = 0\)
- \(\displaystyle 7 \mod 3 = 1\)
- \(\displaystyle 100 \mod 7 = 2\)
Summary: Division Formulas
| Concept | Formula | Description |
|---|---|---|
| Division Algorithm | \(\displaystyle a = b \cdot q + r\) | Fundamental relationship |
| Quotient | \(\displaystyle q = \lfloor a / b \rfloor\) | Integer part of division |
| Remainder | \(\displaystyle r = a - b \cdot q\) | What's left over |
| Modulo | \(\displaystyle a \mod b = r\) | Returns the remainder |
| Divisibility | \(\displaystyle a \mod b = 0\) | \(a\) is divisible by \(b\) |
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