Greatest Common Divisor (GCD) of 105 and 177
The greatest common divisor (GCD) of 105 and 177 is 3.
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 105 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 | 105 ÷ 177 = 0 remainder 105 |
| 2 | 177 ÷ 105 = 1 remainder 72 |
| 3 | 105 ÷ 72 = 1 remainder 33 |
| 4 | 72 ÷ 33 = 2 remainder 6 |
| 5 | 33 ÷ 6 = 5 remainder 3 |
| 6 | 6 ÷ 3 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 125 and 79 | 1 |
| 140 and 149 | 1 |
| 123 and 94 | 1 |
| 136 and 92 | 4 |
| 73 and 112 | 1 |