Greatest Common Divisor (GCD) of 89 and 176
The greatest common divisor (GCD) of 89 and 176 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 89 and 176?
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 | 89 ÷ 176 = 0 remainder 89 |
| 2 | 176 ÷ 89 = 1 remainder 87 |
| 3 | 89 ÷ 87 = 1 remainder 2 |
| 4 | 87 ÷ 2 = 43 remainder 1 |
| 5 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 109 and 11 | 1 |
| 161 and 22 | 1 |
| 11 and 40 | 1 |
| 105 and 23 | 1 |
| 192 and 162 | 6 |