What is GCD(8, 21)?

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

The GCD of 8 and 21 is 1.

How to compute GCD(8, 21)

Comparing the divisors of 8 and 21

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

Divisors of 8:

1, 2, 4, 8

Divisors of 21:

1, 3, 7, 21

We can see from these two lists that the greatest divisor they have in common is: 1

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 21 by 8. The quotient is 2 and the remainder is 5.
  2. The previous divisor (8) is now the dividend. The remainder (5) is the new divisor. Divide 8 by 5. The quotient is 1 and the remainder is 3.
  3. The previous divisor (5) is now the dividend. The remainder (3) is the new divisor. Divide 5 by 3. The quotient is 1 and the remainder is 2.
  4. The previous divisor (3) is now the dividend. The remainder (2) is the new divisor. Divide 3 by 2. The quotient is 1 and the remainder is 1.
  5. The previous divisor (2) is now the dividend. The remainder (1) is the new divisor. Divide 2 by 1. The quotient is 2 and the remainder is 0.
  6. When you reach a remainder of 0, the last divisor (in this case, 1) is the GCD.