Greatest Common Divisor (GCD) of 105 and 86
The greatest common divisor (GCD) of 105 and 86 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 86?
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 ÷ 86 = 1 remainder 19 |
| 2 | 86 ÷ 19 = 4 remainder 10 |
| 3 | 19 ÷ 10 = 1 remainder 9 |
| 4 | 10 ÷ 9 = 1 remainder 1 |
| 5 | 9 ÷ 1 = 9 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 34 and 14 | 2 |
| 117 and 35 | 1 |
| 158 and 165 | 1 |
| 103 and 120 | 1 |
| 57 and 140 | 1 |