Greatest Common Divisor (GCD) of 105 and 171
The greatest common divisor (GCD) of 105 and 171 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 171?
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 ÷ 171 = 0 remainder 105 |
| 2 | 171 ÷ 105 = 1 remainder 66 |
| 3 | 105 ÷ 66 = 1 remainder 39 |
| 4 | 66 ÷ 39 = 1 remainder 27 |
| 5 | 39 ÷ 27 = 1 remainder 12 |
| 6 | 27 ÷ 12 = 2 remainder 3 |
| 7 | 12 ÷ 3 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 136 and 112 | 8 |
| 105 and 177 | 3 |
| 129 and 81 | 3 |
| 27 and 21 | 3 |
| 36 and 47 | 1 |