Greatest Common Divisor (GCD) of 137 and 197
The greatest common divisor (GCD) of 137 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 137 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 | 137 ÷ 197 = 0 remainder 137 |
| 2 | 197 ÷ 137 = 1 remainder 60 |
| 3 | 137 ÷ 60 = 2 remainder 17 |
| 4 | 60 ÷ 17 = 3 remainder 9 |
| 5 | 17 ÷ 9 = 1 remainder 8 |
| 6 | 9 ÷ 8 = 1 remainder 1 |
| 7 | 8 ÷ 1 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 140 and 54 | 2 |
| 42 and 135 | 3 |
| 138 and 90 | 6 |
| 162 and 60 | 6 |
| 97 and 153 | 1 |