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Greatest Common Divisor (GCD) of 84 and 135

The greatest common divisor (GCD) of 84 and 135 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 84 and 135?

We use the Euclidean algorithm, which involves the following steps:

  1. Divide the larger number by the smaller number.
  2. Replace the larger number with the smaller number and the smaller number with the remainder from the division.
  3. Repeat this process until the remainder is zero.
  4. The non-zero remainder just before zero is the GCD.

Step-by-Step Euclidean Algorithm

StepCalculation
1 84 ÷ 135 = 0 remainder 84
2 135 ÷ 84 = 1 remainder 51
3 84 ÷ 51 = 1 remainder 33
4 51 ÷ 33 = 1 remainder 18
5 33 ÷ 18 = 1 remainder 15
6 18 ÷ 15 = 1 remainder 3
7 15 ÷ 3 = 5 remainder 0

Examples of GCD Calculations

NumbersGCD
180 and 1391
104 and 568
37 and 1611
145 and 1805
122 and 982

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