Greatest Common Divisor (GCD) of 45 and 135
The greatest common divisor (GCD) of 45 and 135 is 45.
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 45 and 135?
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 | 45 ÷ 135 = 0 remainder 45 |
| 2 | 135 ÷ 45 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 108 and 87 | 3 |
| 109 and 108 | 1 |
| 59 and 106 | 1 |
| 156 and 111 | 3 |
| 126 and 171 | 9 |