Greatest Common Divisor (GCD) of 65 and 171
The greatest common divisor (GCD) of 65 and 171 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 65 and 171?
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 | 65 ÷ 171 = 0 remainder 65 |
| 2 | 171 ÷ 65 = 2 remainder 41 |
| 3 | 65 ÷ 41 = 1 remainder 24 |
| 4 | 41 ÷ 24 = 1 remainder 17 |
| 5 | 24 ÷ 17 = 1 remainder 7 |
| 6 | 17 ÷ 7 = 2 remainder 3 |
| 7 | 7 ÷ 3 = 2 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 31 and 53 | 1 |
| 61 and 29 | 1 |
| 165 and 14 | 1 |
| 107 and 192 | 1 |
| 159 and 136 | 1 |