Greatest Common Divisor (GCD) of 108 and 175
The greatest common divisor (GCD) of 108 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 108 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 | 108 ÷ 175 = 0 remainder 108 |
| 2 | 175 ÷ 108 = 1 remainder 67 |
| 3 | 108 ÷ 67 = 1 remainder 41 |
| 4 | 67 ÷ 41 = 1 remainder 26 |
| 5 | 41 ÷ 26 = 1 remainder 15 |
| 6 | 26 ÷ 15 = 1 remainder 11 |
| 7 | 15 ÷ 11 = 1 remainder 4 |
| 8 | 11 ÷ 4 = 2 remainder 3 |
| 9 | 4 ÷ 3 = 1 remainder 1 |
| 10 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 166 and 166 | 166 |
| 67 and 170 | 1 |
| 155 and 82 | 1 |
| 39 and 75 | 3 |
| 80 and 153 | 1 |