Greatest Common Divisor (GCD) of 21 and 118
The greatest common divisor (GCD) of 21 and 118 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 21 and 118?
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 | 21 ÷ 118 = 0 remainder 21 |
| 2 | 118 ÷ 21 = 5 remainder 13 |
| 3 | 21 ÷ 13 = 1 remainder 8 |
| 4 | 13 ÷ 8 = 1 remainder 5 |
| 5 | 8 ÷ 5 = 1 remainder 3 |
| 6 | 5 ÷ 3 = 1 remainder 2 |
| 7 | 3 ÷ 2 = 1 remainder 1 |
| 8 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 54 and 51 | 3 |
| 166 and 67 | 1 |
| 136 and 42 | 2 |
| 162 and 16 | 2 |
| 139 and 24 | 1 |