Greatest Common Divisor (GCD) of 122 and 174
The greatest common divisor (GCD) of 122 and 174 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 122 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 | 122 ÷ 174 = 0 remainder 122 |
| 2 | 174 ÷ 122 = 1 remainder 52 |
| 3 | 122 ÷ 52 = 2 remainder 18 |
| 4 | 52 ÷ 18 = 2 remainder 16 |
| 5 | 18 ÷ 16 = 1 remainder 2 |
| 6 | 16 ÷ 2 = 8 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 107 and 174 | 1 |
| 130 and 150 | 10 |
| 17 and 77 | 1 |
| 161 and 164 | 1 |
| 193 and 199 | 1 |