Greatest Common Divisor (GCD) of 105 and 42
The greatest common divisor (GCD) of 105 and 42 is 21.
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 105 and 42?
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 | 105 ÷ 42 = 2 remainder 21 |
| 2 | 42 ÷ 21 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 97 and 74 | 1 |
| 181 and 62 | 1 |
| 82 and 82 | 82 |
| 153 and 83 | 1 |
| 31 and 41 | 1 |