Greatest Common Divisor (GCD) of 137 and 178
The greatest common divisor (GCD) of 137 and 178 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 178?
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 ÷ 178 = 0 remainder 137 |
| 2 | 178 ÷ 137 = 1 remainder 41 |
| 3 | 137 ÷ 41 = 3 remainder 14 |
| 4 | 41 ÷ 14 = 2 remainder 13 |
| 5 | 14 ÷ 13 = 1 remainder 1 |
| 6 | 13 ÷ 1 = 13 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 163 and 60 | 1 |
| 33 and 169 | 1 |
| 169 and 62 | 1 |
| 136 and 85 | 17 |
| 43 and 153 | 1 |