Greatest Common Divisor (GCD) of 111 and 180
The greatest common divisor (GCD) of 111 and 180 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 180?
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 ÷ 180 = 0 remainder 111 |
| 2 | 180 ÷ 111 = 1 remainder 69 |
| 3 | 111 ÷ 69 = 1 remainder 42 |
| 4 | 69 ÷ 42 = 1 remainder 27 |
| 5 | 42 ÷ 27 = 1 remainder 15 |
| 6 | 27 ÷ 15 = 1 remainder 12 |
| 7 | 15 ÷ 12 = 1 remainder 3 |
| 8 | 12 ÷ 3 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 122 and 157 | 1 |
| 96 and 125 | 1 |
| 110 and 144 | 2 |
| 198 and 182 | 2 |
| 145 and 172 | 1 |