What is GCD(65, 90)?

The GCD (Greatest Common Divisor) is the largest number that can divide two (or more) numbers without leaving a remainder.

The GCD of 65 and 90 is 5.

How to compute GCD(65, 90)

Comparing the divisors of 65 and 90

This first method consists in listing the divisors of the two numbers and then identifying the largest one they have in common.

Divisors of 65:

1, 5, 13, 65

Divisors of 90:

1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

We can see from these two lists that the greatest divisor they have in common is: 5

For small numbers, this can be done quickly. However, as numbers increase, the list of potential divisors grows longer, making this method cumbersome and less practical.

Euclid's algorithm

Fortunately, there's a much more efficient method: Euclid's algorithm. It's particularly well-suited to larger numbers. Here's how it works:

  1. Divide 90 by 65. The quotient is 1 and the remainder is 25.
  2. The previous divisor (65) is now the dividend. The remainder (25) is the new divisor. Divide 65 by 25. The quotient is 2 and the remainder is 15.
  3. The previous divisor (25) is now the dividend. The remainder (15) is the new divisor. Divide 25 by 15. The quotient is 1 and the remainder is 10.
  4. The previous divisor (15) is now the dividend. The remainder (10) is the new divisor. Divide 15 by 10. The quotient is 1 and the remainder is 5.
  5. The previous divisor (10) is now the dividend. The remainder (5) is the new divisor. Divide 10 by 5. The quotient is 2 and the remainder is 0.
  6. When you reach a remainder of 0, the last divisor (in this case, 5) is the GCD.