Greatest Common Divisor (GCD) of 106 and 177
The greatest common divisor (GCD) of 106 and 177 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 106 and 177?
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 | 106 ÷ 177 = 0 remainder 106 |
| 2 | 177 ÷ 106 = 1 remainder 71 |
| 3 | 106 ÷ 71 = 1 remainder 35 |
| 4 | 71 ÷ 35 = 2 remainder 1 |
| 5 | 35 ÷ 1 = 35 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 158 and 44 | 2 |
| 196 and 191 | 1 |
| 54 and 141 | 3 |
| 106 and 101 | 1 |
| 45 and 173 | 1 |