Greatest Common Divisor (GCD) of 105 and 142
The greatest common divisor (GCD) of 105 and 142 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 142?
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 ÷ 142 = 0 remainder 105 |
| 2 | 142 ÷ 105 = 1 remainder 37 |
| 3 | 105 ÷ 37 = 2 remainder 31 |
| 4 | 37 ÷ 31 = 1 remainder 6 |
| 5 | 31 ÷ 6 = 5 remainder 1 |
| 6 | 6 ÷ 1 = 6 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 152 and 33 | 1 |
| 151 and 171 | 1 |
| 122 and 96 | 2 |
| 180 and 42 | 6 |
| 49 and 155 | 1 |