Greatest Common Divisor (GCD) of 102 and 180
The greatest common divisor (GCD) of 102 and 180 is 6.
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 102 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 | 102 ÷ 180 = 0 remainder 102 |
| 2 | 180 ÷ 102 = 1 remainder 78 |
| 3 | 102 ÷ 78 = 1 remainder 24 |
| 4 | 78 ÷ 24 = 3 remainder 6 |
| 5 | 24 ÷ 6 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 161 and 124 | 1 |
| 154 and 81 | 1 |
| 36 and 132 | 12 |
| 125 and 159 | 1 |
| 146 and 45 | 1 |