Greatest Common Divisor (GCD) of 65 and 105
The greatest common divisor (GCD) of 65 and 105 is 5.
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 105?
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 ÷ 105 = 0 remainder 65 |
| 2 | 105 ÷ 65 = 1 remainder 40 |
| 3 | 65 ÷ 40 = 1 remainder 25 |
| 4 | 40 ÷ 25 = 1 remainder 15 |
| 5 | 25 ÷ 15 = 1 remainder 10 |
| 6 | 15 ÷ 10 = 1 remainder 5 |
| 7 | 10 ÷ 5 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 67 and 61 | 1 |
| 56 and 22 | 2 |
| 132 and 124 | 4 |
| 82 and 167 | 1 |
| 156 and 36 | 12 |