Greatest Common Divisor (GCD) of 154 and 196
The greatest common divisor (GCD) of 154 and 196 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 154 and 196?
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 | 154 ÷ 196 = 0 remainder 154 |
| 2 | 196 ÷ 154 = 1 remainder 42 |
| 3 | 154 ÷ 42 = 3 remainder 28 |
| 4 | 42 ÷ 28 = 1 remainder 14 |
| 5 | 28 ÷ 14 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 22 and 169 | 1 |
| 113 and 98 | 1 |
| 39 and 160 | 1 |
| 45 and 46 | 1 |
| 120 and 115 | 5 |