Greatest Common Divisor (GCD) of 168 and 105
The greatest common divisor (GCD) of 168 and 105 is 21.
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 168 and 105?
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 | 168 ÷ 105 = 1 remainder 63 |
| 2 | 105 ÷ 63 = 1 remainder 42 |
| 3 | 63 ÷ 42 = 1 remainder 21 |
| 4 | 42 ÷ 21 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 166 and 147 | 1 |
| 45 and 160 | 5 |
| 199 and 185 | 1 |
| 49 and 29 | 1 |
| 126 and 148 | 2 |