Greatest Common Divisor (GCD) of 90 and 109
The greatest common divisor (GCD) of 90 and 109 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 90 and 109?
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 | 90 ÷ 109 = 0 remainder 90 |
| 2 | 109 ÷ 90 = 1 remainder 19 |
| 3 | 90 ÷ 19 = 4 remainder 14 |
| 4 | 19 ÷ 14 = 1 remainder 5 |
| 5 | 14 ÷ 5 = 2 remainder 4 |
| 6 | 5 ÷ 4 = 1 remainder 1 |
| 7 | 4 ÷ 1 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 134 and 94 | 2 |
| 100 and 98 | 2 |
| 168 and 30 | 6 |
| 136 and 198 | 2 |
| 166 and 30 | 2 |