What is the output of the greatest common divisor function gcd(27,12) ?

gcd(x,y)={x,gcd(y,x%y),y=0y>0 where x%y is remainder division or modulo

(1 point)
Responses

12
12

3
3

27
27

1

1 answer

To find the greatest common divisor (gcd) of 27 and 12 using the provided function, we can apply the Euclidean algorithm.

  1. First, we calculate \( 27 \mod 12 \):

    • \( 27 \div 12 = 2 \) (which gives us a quotient of 2)
    • \( 2 \times 12 = 24 \)
    • \( 27 - 24 = 3 \)
    • Thus, \( 27 \mod 12 = 3 \).
  2. Now we apply the gcd function:

    • \( \text{gcd}(27, 12) = \text{gcd}(12, 3) \).
  3. Next, we calculate \( 12 \mod 3 \):

    • \( 12 \div 3 = 4 \) (which gives us a quotient of 4)
    • \( 4 \times 3 = 12 \)
    • \( 12 - 12 = 0 \)
    • Thus, \( 12 \mod 3 = 0 \).
  4. Now we apply the gcd function again:

    • \( \text{gcd}(12, 3) = \text{gcd}(3, 0) \).
  5. The gcd of any number and 0 is that number itself:

    • \( \text{gcd}(3, 0) = 3 \).

Thus, the output of \( \text{gcd}(27, 12) \) is 3.

The correct response is: 3