Greatest Common Divisor (GCD) of 65 and 93
The greatest common divisor (GCD) of 65 and 93 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 93?
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 ÷ 93 = 0 remainder 65 |
| 2 | 93 ÷ 65 = 1 remainder 28 |
| 3 | 65 ÷ 28 = 2 remainder 9 |
| 4 | 28 ÷ 9 = 3 remainder 1 |
| 5 | 9 ÷ 1 = 9 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 46 and 170 | 2 |
| 169 and 101 | 1 |
| 71 and 76 | 1 |
| 175 and 182 | 7 |
| 106 and 98 | 2 |