Greatest Common Divisor (GCD) of 82 and 114
The greatest common divisor (GCD) of 82 and 114 is 2.
What is the Greatest Common Divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is useful in simplifying fractions, finding common factors, and in number theory.
How to Calculate the GCD of 82 and 114?
We use the Euclidean algorithm, which involves the following steps:
- Divide the larger number by the smaller number.
- Replace the larger number with the smaller number and the smaller number with the remainder from the division.
- Repeat this process until the remainder is zero.
- The non-zero remainder just before zero is the GCD.
Step-by-Step Euclidean Algorithm
| Step | Calculation |
|---|---|
| 1 | 82 ÷ 114 = 0 remainder 82 |
| 2 | 114 ÷ 82 = 1 remainder 32 |
| 3 | 82 ÷ 32 = 2 remainder 18 |
| 4 | 32 ÷ 18 = 1 remainder 14 |
| 5 | 18 ÷ 14 = 1 remainder 4 |
| 6 | 14 ÷ 4 = 3 remainder 2 |
| 7 | 4 ÷ 2 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 113 and 63 | 1 |
| 11 and 168 | 1 |
| 183 and 13 | 1 |
| 113 and 142 | 1 |
| 112 and 191 | 1 |