Greatest Common Divisor (GCD) of 172 and 67
The greatest common divisor (GCD) of 172 and 67 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 172 and 67?
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 | 172 ÷ 67 = 2 remainder 38 |
| 2 | 67 ÷ 38 = 1 remainder 29 |
| 3 | 38 ÷ 29 = 1 remainder 9 |
| 4 | 29 ÷ 9 = 3 remainder 2 |
| 5 | 9 ÷ 2 = 4 remainder 1 |
| 6 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 21 and 188 | 1 |
| 195 and 120 | 15 |
| 141 and 116 | 1 |
| 121 and 106 | 1 |
| 81 and 82 | 1 |