What is GCD(67, 84)?

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

The GCD of 67 and 84 is 1.

How to compute GCD(67, 84)

Comparing the divisors of 67 and 84

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

Divisors of 67:

1, 67

Divisors of 84:

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

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 84 by 67. The quotient is 1 and the remainder is 17.
  2. The previous divisor (67) is now the dividend. The remainder (17) is the new divisor. Divide 67 by 17. The quotient is 3 and the remainder is 16.
  3. The previous divisor (17) is now the dividend. The remainder (16) is the new divisor. Divide 17 by 16. The quotient is 1 and the remainder is 1.
  4. The previous divisor (16) is now the dividend. The remainder (1) is the new divisor. Divide 16 by 1. The quotient is 16 and the remainder is 0.
  5. When you reach a remainder of 0, the last divisor (in this case, 1) is the GCD.