Greatest Common Divisor (GCD) of 105 and 156
The greatest common divisor (GCD) of 105 and 156 is 3.
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 156?
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 ÷ 156 = 0 remainder 105 |
| 2 | 156 ÷ 105 = 1 remainder 51 |
| 3 | 105 ÷ 51 = 2 remainder 3 |
| 4 | 51 ÷ 3 = 17 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 140 and 24 | 4 |
| 136 and 193 | 1 |
| 20 and 77 | 1 |
| 141 and 122 | 1 |
| 139 and 76 | 1 |