Greatest Common Divisor (GCD) of 84 and 157
The greatest common divisor (GCD) of 84 and 157 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 84 and 157?
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 | 84 ÷ 157 = 0 remainder 84 |
| 2 | 157 ÷ 84 = 1 remainder 73 |
| 3 | 84 ÷ 73 = 1 remainder 11 |
| 4 | 73 ÷ 11 = 6 remainder 7 |
| 5 | 11 ÷ 7 = 1 remainder 4 |
| 6 | 7 ÷ 4 = 1 remainder 3 |
| 7 | 4 ÷ 3 = 1 remainder 1 |
| 8 | 3 ÷ 1 = 3 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 131 and 44 | 1 |
| 141 and 75 | 3 |
| 116 and 164 | 4 |
| 55 and 110 | 55 |
| 114 and 157 | 1 |