Greatest Common Divisor (GCD) of 85 and 135
The greatest common divisor (GCD) of 85 and 135 is 5.
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 85 and 135?
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 | 85 ÷ 135 = 0 remainder 85 |
| 2 | 135 ÷ 85 = 1 remainder 50 |
| 3 | 85 ÷ 50 = 1 remainder 35 |
| 4 | 50 ÷ 35 = 1 remainder 15 |
| 5 | 35 ÷ 15 = 2 remainder 5 |
| 6 | 15 ÷ 5 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 198 and 172 | 2 |
| 179 and 96 | 1 |
| 135 and 56 | 1 |
| 160 and 82 | 2 |
| 153 and 125 | 1 |