Final answer:
To find the GCD of 90 and 54, the Euclidean algorithm involves dividing the larger number by the smaller, then continuing the process with the divisor and the remainder. This continues until the remainder is zero; the last nonzero remainder is the GCD, which in this case is 18.
Step-by-step explanation:
To find the greatest common divisor (GCD) of 90 and 54 using the Euclidean algorithm, you follow a series of steps where you divide the larger number by the smaller and then replace the larger number with the smaller and the smaller number with the remainder from the division process. Here is the work shown:Divide 90 by 54, which goes 1 time with a remainder of 36. So, 90 = 54 × 1 + 36.
Now, take 54 and divide it by the remainder, 36. This goes 1 time with a remainder of 18. So, 54 = 36 × 1 + 18.
Next, divide 36 by 18, which goes exactly 2 times with no remainder. So, 36 = 18 × 2 + 0.
Since the remainder is now 0, the GCD is the last nonzero remainder, which is 18.