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Greatest Common Divisor (GCD) of 180 and 105

The greatest common divisor (GCD) of 180 and 105 is 15.

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 180 and 105?

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
1 180 ÷ 105 = 1 remainder 75
2 105 ÷ 75 = 1 remainder 30
3 75 ÷ 30 = 2 remainder 15
4 30 ÷ 15 = 2 remainder 0

Examples of GCD Calculations

NumbersGCD
112 and 1197
92 and 1182
75 and 1091
47 and 691
95 and 1391

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