Greatest Common Divisor (GCD) of 109 and 144
The greatest common divisor (GCD) of 109 and 144 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 109 and 144?
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 | 109 ÷ 144 = 0 remainder 109 |
| 2 | 144 ÷ 109 = 1 remainder 35 |
| 3 | 109 ÷ 35 = 3 remainder 4 |
| 4 | 35 ÷ 4 = 8 remainder 3 |
| 5 | 4 ÷ 3 = 1 remainder 1 |
| 6 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 112 and 135 | 1 |
| 131 and 73 | 1 |
| 164 and 18 | 2 |
| 163 and 18 | 1 |
| 172 and 58 | 2 |