Barney Is Forming Two Large Cubes, A And B, With Small Identical Cubes GMAT Problem Solving

Question: Barney is forming two large cubes, A and B, with small identical cubes. He used x small cubes to form cube A and y small cubes to form cube B. If x - y = 61, then what is the value of x + y?

1. 64
2. 125
3. 169
4. 189
5. 216

“Barney is forming two large cubes, A and B, with small identical cubes” - is a topic that is covered in the quantitative reasoning section of the GMAT. To successfully execute the GMAT Problem Solving questions, a student must possess a wide range of qualitative skills. The entire GMAT Quant section consists of 31 questions. The problem-solving section of the GMAT Quant topics requires the solution of calculative mathematical problems.

Solutions and Explanation

Approach Solution : 1

Due to the fact that large cubes are constructed from small, identical cubes, x and y must both be perfect cubes (a perfect cube is a cube of an integer).
Therefore, x and y could be any of the following: 1, 8, 27, 64, 125, 216,…
Given that x-y=61, So we can interpret that x and y are respectively 125 and 64.

Hence,
x+y= 125+64
Therefore, x+y = 189

Correct Answer: (D)

Approach Solution : 2

The numbers x and y must be perfect cubes in order to form a cube.

Let x = a^3 and y = b^3 be equal.
Given that Given a^3 − b^3 =61
(a−b)(a2+ab+b2)=61
Given that 61 is a prime number and the above equation is a product of two integers, we can draw the following conclusion.
(a−b)=1 meaning that a=b+1 and (a2+ab+b2)=61

Replace the above a value in the question
we get (b+1)^2 + (b+1)* b+ b^2=61
b^2 + 2b + 1+ b^2 +b + b^2 = 61
3b^2 + 3^b + 1 = 61
b^2 + b−20 = 0

As a result, b = 4 or -5

since b should be positive
We get b = 4 and a = 5, where x = a^3 = 125 and y = b^3 = 64.
Consequently, x+y = 125 + 64 = 189.

Correct Answer: (D)

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