Greatest Common Divisor (GCD) of 164 and 65
The greatest common divisor (GCD) of 164 and 65 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 164 and 65?
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 | 164 ÷ 65 = 2 remainder 34 |
| 2 | 65 ÷ 34 = 1 remainder 31 |
| 3 | 34 ÷ 31 = 1 remainder 3 |
| 4 | 31 ÷ 3 = 10 remainder 1 |
| 5 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 129 and 49 | 1 |
| 178 and 129 | 1 |
| 161 and 187 | 1 |
| 29 and 160 | 1 |
| 31 and 19 | 1 |