Greatest Common Divisor (GCD) of 112 and 154
The greatest common divisor (GCD) of 112 and 154 is 14.
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 112 and 154?
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 | 112 ÷ 154 = 0 remainder 112 |
| 2 | 154 ÷ 112 = 1 remainder 42 |
| 3 | 112 ÷ 42 = 2 remainder 28 |
| 4 | 42 ÷ 28 = 1 remainder 14 |
| 5 | 28 ÷ 14 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 101 and 182 | 1 |
| 65 and 79 | 1 |
| 158 and 191 | 1 |
| 163 and 200 | 1 |
| 176 and 47 | 1 |