Florida Teacher Certification Examinations (FTCE) Subject Area Practice Test

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What is the greatest common factor (GCF) of the numbers 36, 135, and 144?

The correct answer is: 9

To determine the greatest common factor (GCF) of the numbers 36, 135, and 144, it is essential to analyze the prime factorization of each number first. Starting with 36, its prime factorization is: - 36 = 2 × 2 × 3 × 3 = $2^2 \times 3^2$ Next, for 135, the prime factorization is: - 135 = 3 × 3 × 3 × 5 = $3^3 \times 5^1$ Lastly, for 144, the prime factorization is: - 144 = 2 × 2 × 2 × 3 × 3 = $2^3 \times 3^2$ Now, to find the GCF, we identify the common prime factors in all three factorizations and take the lowest power of each common factor: - The prime factor 2 appears in 36 and 144 but not in 135, so it is not included in the GCF. - The prime factor 3 appears in all three numbers. The lowest power of 3 in the factorizations is $3^2$ (from