Greatest Common Divisor (GCD) of 113 and 164
The greatest common divisor (GCD) of 113 and 164 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 164?
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 ÷ 164 = 0 remainder 113 |
| 2 | 164 ÷ 113 = 1 remainder 51 |
| 3 | 113 ÷ 51 = 2 remainder 11 |
| 4 | 51 ÷ 11 = 4 remainder 7 |
| 5 | 11 ÷ 7 = 1 remainder 4 |
| 6 | 7 ÷ 4 = 1 remainder 3 |
| 7 | 4 ÷ 3 = 1 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 38 and 38 | 38 |
| 89 and 37 | 1 |
| 24 and 108 | 12 |
| 154 and 132 | 22 |
| 199 and 113 | 1 |