Greatest Common Divisor (GCD) of 109 and 54
The greatest common divisor (GCD) of 109 and 54 is 1.
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 109 and 54?
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 | 109 ÷ 54 = 2 remainder 1 |
| 2 | 54 ÷ 1 = 54 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 197 and 80 | 1 |
| 15 and 133 | 1 |
| 155 and 150 | 5 |
| 149 and 92 | 1 |
| 185 and 55 | 5 |