Greatest Common Divisor (GCD) of 105 and 180
The greatest common divisor (GCD) of 105 and 180 is 15.
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 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 | 105 ÷ 180 = 0 remainder 105 |
| 2 | 180 ÷ 105 = 1 remainder 75 |
| 3 | 105 ÷ 75 = 1 remainder 30 |
| 4 | 75 ÷ 30 = 2 remainder 15 |
| 5 | 30 ÷ 15 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 45 and 15 | 15 |
| 115 and 80 | 5 |
| 26 and 102 | 2 |
| 85 and 100 | 5 |
| 176 and 80 | 16 |