Greatest Common Divisor (GCD) of 113 and 176
The greatest common divisor (GCD) of 113 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 113 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 | 113 ÷ 176 = 0 remainder 113 |
| 2 | 176 ÷ 113 = 1 remainder 63 |
| 3 | 113 ÷ 63 = 1 remainder 50 |
| 4 | 63 ÷ 50 = 1 remainder 13 |
| 5 | 50 ÷ 13 = 3 remainder 11 |
| 6 | 13 ÷ 11 = 1 remainder 2 |
| 7 | 11 ÷ 2 = 5 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 116 and 41 | 1 |
| 122 and 89 | 1 |
| 189 and 80 | 1 |
| 156 and 75 | 3 |
| 93 and 82 | 1 |