Greatest Common Divisor (GCD) of 109 and 134
The greatest common divisor (GCD) of 109 and 134 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 134?
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 ÷ 134 = 0 remainder 109 |
| 2 | 134 ÷ 109 = 1 remainder 25 |
| 3 | 109 ÷ 25 = 4 remainder 9 |
| 4 | 25 ÷ 9 = 2 remainder 7 |
| 5 | 9 ÷ 7 = 1 remainder 2 |
| 6 | 7 ÷ 2 = 3 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 158 and 35 | 1 |
| 142 and 83 | 1 |
| 149 and 136 | 1 |
| 115 and 80 | 5 |
| 176 and 43 | 1 |