Greatest Common Divisor (GCD) of 137 and 192
The greatest common divisor (GCD) of 137 and 192 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 192?
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 ÷ 192 = 0 remainder 137 |
| 2 | 192 ÷ 137 = 1 remainder 55 |
| 3 | 137 ÷ 55 = 2 remainder 27 |
| 4 | 55 ÷ 27 = 2 remainder 1 |
| 5 | 27 ÷ 1 = 27 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 189 and 153 | 9 |
| 166 and 175 | 1 |
| 37 and 22 | 1 |
| 185 and 128 | 1 |
| 188 and 109 | 1 |