Greatest Common Divisor (GCD) of 105 and 71
The greatest common divisor (GCD) of 105 and 71 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 105 and 71?
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 ÷ 71 = 1 remainder 34 |
| 2 | 71 ÷ 34 = 2 remainder 3 |
| 3 | 34 ÷ 3 = 11 remainder 1 |
| 4 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 131 and 199 | 1 |
| 109 and 138 | 1 |
| 100 and 154 | 2 |
| 170 and 43 | 1 |
| 100 and 148 | 4 |