Greatest Common Divisor (GCD) of 109 and 175
The greatest common divisor (GCD) of 109 and 175 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 175?
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 ÷ 175 = 0 remainder 109 |
| 2 | 175 ÷ 109 = 1 remainder 66 |
| 3 | 109 ÷ 66 = 1 remainder 43 |
| 4 | 66 ÷ 43 = 1 remainder 23 |
| 5 | 43 ÷ 23 = 1 remainder 20 |
| 6 | 23 ÷ 20 = 1 remainder 3 |
| 7 | 20 ÷ 3 = 6 remainder 2 |
| 8 | 3 ÷ 2 = 1 remainder 1 |
| 9 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 159 and 185 | 1 |
| 84 and 42 | 42 |
| 200 and 79 | 1 |
| 115 and 90 | 5 |
| 180 and 88 | 4 |