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Nov 30, 2024

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If x represents the total number of positive integers that are factors of 240, and y represents the total number of positive integers that are factors of 120, what is the value of x2 y2?

0

2

12

144

256

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Scott Woodbury-Stewart Founder and Expert GMAT Instructor

Solution:

Begin by reducing 240 to its prime factors.

240=24×10 240=6×4×2×5 240=2×3×2×2×2×5 240= 2 4 × 3 1 × 5 1

Next, add 1 to each of the exponents and multiply the results: (4 + 1)(1 + 1)(1 + 1) = (5)(2)(2) = 20. Thus, 240 has a total of 20 factors, which means x = 20.

Next, let’s find the prime factorization of 120.

120=12×10 120=4×3×2×5 120=2×2×3×2×5 120= 2 3 × 3 1 × 5 1

After adding 1 to each exponent and multiplying the results, we get (3 + 1)(1 + 1)(1 + 1) = (4)(2)(2) = 16. Thus, 120 has 16 different factors, which means y = 16.

Finally, we need to determine x2 y2:

x 2 y 2 = 20 2 16 2 =400256=144

Correct answer: D
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