Greatest Common Divisor (GCD) of 111 and 178
The greatest common divisor (GCD) of 111 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 111 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 | 111 ÷ 178 = 0 remainder 111 |
| 2 | 178 ÷ 111 = 1 remainder 67 |
| 3 | 111 ÷ 67 = 1 remainder 44 |
| 4 | 67 ÷ 44 = 1 remainder 23 |
| 5 | 44 ÷ 23 = 1 remainder 21 |
| 6 | 23 ÷ 21 = 1 remainder 2 |
| 7 | 21 ÷ 2 = 10 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 63 and 50 | 1 |
| 21 and 182 | 7 |
| 125 and 153 | 1 |
| 13 and 70 | 1 |
| 136 and 174 | 2 |