Greatest Common Divisor (GCD) of 133 and 84
The greatest common divisor (GCD) of 133 and 84 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 133 and 84?
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 | 133 ÷ 84 = 1 remainder 49 |
| 2 | 84 ÷ 49 = 1 remainder 35 |
| 3 | 49 ÷ 35 = 1 remainder 14 |
| 4 | 35 ÷ 14 = 2 remainder 7 |
| 5 | 14 ÷ 7 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 54 and 71 | 1 |
| 115 and 151 | 1 |
| 145 and 140 | 5 |
| 89 and 12 | 1 |
| 114 and 123 | 3 |