Greatest Common Divisor (GCD) of 137 and 163
The greatest common divisor (GCD) of 137 and 163 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 163?
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 ÷ 163 = 0 remainder 137 |
| 2 | 163 ÷ 137 = 1 remainder 26 |
| 3 | 137 ÷ 26 = 5 remainder 7 |
| 4 | 26 ÷ 7 = 3 remainder 5 |
| 5 | 7 ÷ 5 = 1 remainder 2 |
| 6 | 5 ÷ 2 = 2 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 164 and 41 | 41 |
| 137 and 144 | 1 |
| 109 and 37 | 1 |
| 92 and 32 | 4 |
| 184 and 107 | 1 |