Greatest Common Divisor (GCD) of 65 and 172
The greatest common divisor (GCD) of 65 and 172 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 65 and 172?
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 | 65 ÷ 172 = 0 remainder 65 |
| 2 | 172 ÷ 65 = 2 remainder 42 |
| 3 | 65 ÷ 42 = 1 remainder 23 |
| 4 | 42 ÷ 23 = 1 remainder 19 |
| 5 | 23 ÷ 19 = 1 remainder 4 |
| 6 | 19 ÷ 4 = 4 remainder 3 |
| 7 | 4 ÷ 3 = 1 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 180 and 194 | 2 |
| 174 and 15 | 3 |
| 80 and 199 | 1 |
| 125 and 64 | 1 |
| 72 and 134 | 2 |