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Greatest Common Divisor (GCD) of 65 and 45

The greatest common divisor (GCD) of 65 and 45 is 5.

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 65 and 45?

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 65 ÷ 45 = 1 remainder 20
2 45 ÷ 20 = 2 remainder 5
3 20 ÷ 5 = 4 remainder 0

Examples of GCD Calculations

NumbersGCD
51 and 1641
123 and 711
200 and 1371
148 and 1011
87 and 1461

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