Greatest Common Divisor (GCD) of 111 and 66
The greatest common divisor (GCD) of 111 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 111 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 | 111 ÷ 66 = 1 remainder 45 |
| 2 | 66 ÷ 45 = 1 remainder 21 |
| 3 | 45 ÷ 21 = 2 remainder 3 |
| 4 | 21 ÷ 3 = 7 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 143 and 193 | 1 |
| 128 and 38 | 2 |
| 158 and 167 | 1 |
| 179 and 177 | 1 |
| 179 and 74 | 1 |