Greatest Common Divisor (GCD) of 114 and 197
The greatest common divisor (GCD) of 114 and 197 is 1.
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 114 and 197?
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 | 114 ÷ 197 = 0 remainder 114 |
| 2 | 197 ÷ 114 = 1 remainder 83 |
| 3 | 114 ÷ 83 = 1 remainder 31 |
| 4 | 83 ÷ 31 = 2 remainder 21 |
| 5 | 31 ÷ 21 = 1 remainder 10 |
| 6 | 21 ÷ 10 = 2 remainder 1 |
| 7 | 10 ÷ 1 = 10 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 194 and 188 | 2 |
| 191 and 149 | 1 |
| 103 and 17 | 1 |
| 152 and 67 | 1 |
| 82 and 37 | 1 |