Final answer:
To find the greatest common divisor (gcd) of 27648 and 450 using the Euclidean algorithm, we divide the larger number by the smaller number and find the remainder. We repeat this process until the remainder is zero. The gcd is then the last non-zero remainder.
Step-by-step explanation:
First, we divide 27648 by 450 and find the quotient and remainder: 27648 ÷ 450 = 61 with a remainder of 198. Then, we divide 450 by 198 and find the quotient and remainder: 450 ÷ 198 = 2 with a remainder of 54. Continuing this process, we divide 198 by 54 and find the quotient and remainder: 198 ÷ 54 = 3 with a remainder of 36. Finally, we divide 54 by 36 and find the quotient and remainder: 54 ÷ 36 = 1 with a remainder of 18. Since the remainder in the last step is not zero, we repeat the process using the divisor (36) and the last non-zero remainder (18). We divide 36 by 18 and find the quotient and remainder: 36 ÷ 18 = 2 with a remainder of 0. Since the remainder is now zero, the greatest common divisor (gcd) of 27648 and 450 is the last non-zero remainder, which is 18. Therefore, ( operatorname{gcd}(27648,450) ) = 18.