Greatest Common Divisor (GCD) of 109 and 155
The greatest common divisor (GCD) of 109 and 155 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 155?
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 ÷ 155 = 0 remainder 109 |
| 2 | 155 ÷ 109 = 1 remainder 46 |
| 3 | 109 ÷ 46 = 2 remainder 17 |
| 4 | 46 ÷ 17 = 2 remainder 12 |
| 5 | 17 ÷ 12 = 1 remainder 5 |
| 6 | 12 ÷ 5 = 2 remainder 2 |
| 7 | 5 ÷ 2 = 2 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 109 and 78 | 1 |
| 167 and 98 | 1 |
| 162 and 108 | 54 |
| 177 and 16 | 1 |
| 160 and 174 | 2 |