Math  /  Numbers & Operations

QuestionWhat is the GCF of 30 and 54? 3 6 9 15

Studdy Solution

STEP 1

1. The problem requires finding the Greatest Common Factor (GCF) of two numbers, 30 and 54.
2. The GCF is the largest number that divides both 30 and 54 without leaving a remainder.
3. We can use the Euclidean algorithm to find the GCF.

STEP 2

1. Apply the Euclidean algorithm to find the GCF of 30 and 54.

STEP 3

Apply the Euclidean algorithm, which states that the GCF of two numbers aa and bb can be found by repeatedly applying the operation amodba \mod b until bb becomes zero. The GCF is the last non-zero remainder.
First, compute 54mod3054 \mod 30: 54mod30=5430×1=24 54 \mod 30 = 54 - 30 \times 1 = 24

STEP 4

Next, compute 30mod2430 \mod 24: 30mod24=3024×1=6 30 \mod 24 = 30 - 24 \times 1 = 6

STEP 5

Then, compute 24mod624 \mod 6: 24mod6=246×4=0 24 \mod 6 = 24 - 6 \times 4 = 0

STEP 6

Since the remainder is now zero, the last non-zero remainder is the GCF. Thus, the GCF of 30 and 54 is: GCF=6 \text{GCF} = 6

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