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Greatest Common Divisor (GCD) of 105 and 64

The greatest common divisor (GCD) of 105 and 64 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 105 and 64?

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
1 105 ÷ 64 = 1 remainder 41
2 64 ÷ 41 = 1 remainder 23
3 41 ÷ 23 = 1 remainder 18
4 23 ÷ 18 = 1 remainder 5
5 18 ÷ 5 = 3 remainder 3
6 5 ÷ 3 = 1 remainder 2
7 3 ÷ 2 = 1 remainder 1
8 2 ÷ 1 = 2 remainder 0

Examples of GCD Calculations

NumbersGCD
115 and 1391
150 and 1293
102 and 1146
117 and 101
45 and 1041

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