Greatest Common Divisor (GCD) of 105 and 66
The greatest common divisor (GCD) of 105 and 66 is 3.
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 105 and 66?
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 | 105 ÷ 66 = 1 remainder 39 |
| 2 | 66 ÷ 39 = 1 remainder 27 |
| 3 | 39 ÷ 27 = 1 remainder 12 |
| 4 | 27 ÷ 12 = 2 remainder 3 |
| 5 | 12 ÷ 3 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 186 and 89 | 1 |
| 98 and 192 | 2 |
| 111 and 101 | 1 |
| 18 and 88 | 2 |
| 127 and 103 | 1 |