Greatest Common Divisor (GCD) of 127 and 197
The greatest common divisor (GCD) of 127 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 127 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 | 127 ÷ 197 = 0 remainder 127 |
| 2 | 197 ÷ 127 = 1 remainder 70 |
| 3 | 127 ÷ 70 = 1 remainder 57 |
| 4 | 70 ÷ 57 = 1 remainder 13 |
| 5 | 57 ÷ 13 = 4 remainder 5 |
| 6 | 13 ÷ 5 = 2 remainder 3 |
| 7 | 5 ÷ 3 = 1 remainder 2 |
| 8 | 3 ÷ 2 = 1 remainder 1 |
| 9 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 89 and 119 | 1 |
| 120 and 172 | 4 |
| 172 and 190 | 2 |
| 135 and 143 | 1 |
| 105 and 160 | 5 |