What is GCD(54, 69)?

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

The GCD of 54 and 69 is 3.

How to compute GCD(54, 69)

Comparing the divisors of 54 and 69

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

Divisors of 54:

1, 2, 3, 6, 9, 18, 27, 54

Divisors of 69:

1, 3, 23, 69

We can see from these two lists that the greatest divisor they have in common is: 3

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 69 by 54. The quotient is 1 and the remainder is 15.
  2. The previous divisor (54) is now the dividend. The remainder (15) is the new divisor. Divide 54 by 15. The quotient is 3 and the remainder is 9.
  3. The previous divisor (15) is now the dividend. The remainder (9) is the new divisor. Divide 15 by 9. The quotient is 1 and the remainder is 6.
  4. The previous divisor (9) is now the dividend. The remainder (6) is the new divisor. Divide 9 by 6. The quotient is 1 and the remainder is 3.
  5. The previous divisor (6) is now the dividend. The remainder (3) is the new divisor. Divide 6 by 3. The quotient is 2 and the remainder is 0.
  6. When you reach a remainder of 0, the last divisor (in this case, 3) is the GCD.