Greatest Common Divisor (GCD) of 109 and 178
The greatest common divisor (GCD) of 109 and 178 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 178?
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 ÷ 178 = 0 remainder 109 |
| 2 | 178 ÷ 109 = 1 remainder 69 |
| 3 | 109 ÷ 69 = 1 remainder 40 |
| 4 | 69 ÷ 40 = 1 remainder 29 |
| 5 | 40 ÷ 29 = 1 remainder 11 |
| 6 | 29 ÷ 11 = 2 remainder 7 |
| 7 | 11 ÷ 7 = 1 remainder 4 |
| 8 | 7 ÷ 4 = 1 remainder 3 |
| 9 | 4 ÷ 3 = 1 remainder 1 |
| 10 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 10 and 194 | 2 |
| 154 and 14 | 14 |
| 119 and 46 | 1 |
| 86 and 122 | 2 |
| 10 and 165 | 5 |