Greatest Common Divisor (GCD) of 67 and 38
The greatest common divisor (GCD) of 67 and 38 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 67 and 38?
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 | 67 ÷ 38 = 1 remainder 29 |
| 2 | 38 ÷ 29 = 1 remainder 9 |
| 3 | 29 ÷ 9 = 3 remainder 2 |
| 4 | 9 ÷ 2 = 4 remainder 1 |
| 5 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 38 and 34 | 2 |
| 125 and 164 | 1 |
| 114 and 56 | 2 |
| 127 and 117 | 1 |
| 113 and 184 | 1 |