Greatest Common Divisor (GCD) of 65 and 169
The greatest common divisor (GCD) of 65 and 169 is 13.
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 169?
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 ÷ 169 = 0 remainder 65 |
| 2 | 169 ÷ 65 = 2 remainder 39 |
| 3 | 65 ÷ 39 = 1 remainder 26 |
| 4 | 39 ÷ 26 = 1 remainder 13 |
| 5 | 26 ÷ 13 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 141 and 148 | 1 |
| 80 and 176 | 16 |
| 67 and 69 | 1 |
| 186 and 28 | 2 |
| 55 and 62 | 1 |