Greatest Common Divisor (GCD) of 105 and 174
The greatest common divisor (GCD) of 105 and 174 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 174?
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 ÷ 174 = 0 remainder 105 |
| 2 | 174 ÷ 105 = 1 remainder 69 |
| 3 | 105 ÷ 69 = 1 remainder 36 |
| 4 | 69 ÷ 36 = 1 remainder 33 |
| 5 | 36 ÷ 33 = 1 remainder 3 |
| 6 | 33 ÷ 3 = 11 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 103 and 119 | 1 |
| 200 and 168 | 8 |
| 86 and 71 | 1 |
| 156 and 168 | 12 |
| 107 and 198 | 1 |