Greatest Common Divisor (GCD) of 172 and 106
The greatest common divisor (GCD) of 172 and 106 is 2.
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 106?
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 ÷ 106 = 1 remainder 66 |
| 2 | 106 ÷ 66 = 1 remainder 40 |
| 3 | 66 ÷ 40 = 1 remainder 26 |
| 4 | 40 ÷ 26 = 1 remainder 14 |
| 5 | 26 ÷ 14 = 1 remainder 12 |
| 6 | 14 ÷ 12 = 1 remainder 2 |
| 7 | 12 ÷ 2 = 6 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 197 and 172 | 1 |
| 25 and 62 | 1 |
| 26 and 105 | 1 |
| 40 and 51 | 1 |
| 159 and 139 | 1 |