Greatest Common Divisor (GCD) of 105 and 137
The greatest common divisor (GCD) of 105 and 137 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 105 and 137?
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 | 105 ÷ 137 = 0 remainder 105 |
| 2 | 137 ÷ 105 = 1 remainder 32 |
| 3 | 105 ÷ 32 = 3 remainder 9 |
| 4 | 32 ÷ 9 = 3 remainder 5 |
| 5 | 9 ÷ 5 = 1 remainder 4 |
| 6 | 5 ÷ 4 = 1 remainder 1 |
| 7 | 4 ÷ 1 = 4 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 55 and 50 | 5 |
| 44 and 189 | 1 |
| 186 and 42 | 6 |
| 185 and 101 | 1 |
| 101 and 141 | 1 |