Greatest Common Divisor (GCD) of 109 and 174
The greatest common divisor (GCD) of 109 and 174 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 109 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 | 109 ÷ 174 = 0 remainder 109 |
| 2 | 174 ÷ 109 = 1 remainder 65 |
| 3 | 109 ÷ 65 = 1 remainder 44 |
| 4 | 65 ÷ 44 = 1 remainder 21 |
| 5 | 44 ÷ 21 = 2 remainder 2 |
| 6 | 21 ÷ 2 = 10 remainder 1 |
| 7 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 189 and 109 | 1 |
| 25 and 175 | 25 |
| 93 and 87 | 3 |
| 45 and 139 | 1 |
| 149 and 170 | 1 |