Greatest Common Divisor (GCD) of 84 and 119
The greatest common divisor (GCD) of 84 and 119 is 7.
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 84 and 119?
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 | 84 ÷ 119 = 0 remainder 84 |
| 2 | 119 ÷ 84 = 1 remainder 35 |
| 3 | 84 ÷ 35 = 2 remainder 14 |
| 4 | 35 ÷ 14 = 2 remainder 7 |
| 5 | 14 ÷ 7 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 182 and 115 | 1 |
| 112 and 143 | 1 |
| 63 and 155 | 1 |
| 195 and 75 | 15 |
| 52 and 178 | 2 |