Greatest Common Divisor (GCD) of 38 and 106
The greatest common divisor (GCD) of 38 and 106 is 2.
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 38 and 106?
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 | 38 ÷ 106 = 0 remainder 38 |
| 2 | 106 ÷ 38 = 2 remainder 30 |
| 3 | 38 ÷ 30 = 1 remainder 8 |
| 4 | 30 ÷ 8 = 3 remainder 6 |
| 5 | 8 ÷ 6 = 1 remainder 2 |
| 6 | 6 ÷ 2 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 197 and 98 | 1 |
| 105 and 138 | 3 |
| 68 and 24 | 4 |
| 149 and 41 | 1 |
| 170 and 47 | 1 |