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Bunuel

Fresh GMAT Club Tests' 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?

A. 64
B. 125
C. 169
D. 189
E. 216

Check other 3-D Geometry Questions in our Special Questions Directory.
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Bunuel

Fresh GMAT Club Tests' 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?

A. 64
B. 125
C. 169
D. 189
E. 216

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

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Given x-y = 61.

Lets try out the options.

From 1) x+y = 64.
Then 2x = 125.
x has to be an integer. Hence 1 is wrong.

From 2) x+y = 125.
Then 2x = 186.
x = 93 and y = 32.
Since x and y are cubes. Using 93 smaller cubes, a bigger cube cannot be made. Hence option 2 is wrong.

Similarly checking the rest of the options. From 4, we get x = 125 and y = 64.

x = 125 = 5*5*5
y = 64 = 4*4*4.

Hence D is the correct option

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Bunuel

Fresh GMAT Club Tests' 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?

A. 64
B. 125
C. 169
D. 189
E. 216

If a large cube is made of small cubes, then the number of small cubes is the cube of a side of the large cube;
For instance: Large cube with side 2 can be made of 2^3=8 smaller cubes;

so x-y=61 is the difference between two perfect cubes;
perfect cubes: [1,8,27,64,125,216,343…];
x=125, y=64, x-y=61… x+y=169

Ans (C)
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As the large cubes are made up of smaller cubes, the number of smaller cubes x and y must be one of these integers:
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
from here we see that x-y = 125-64 = 61 and x+y = 125+64 = 189.
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Bunuel

Fresh GMAT Club Tests' 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?

A. 64
B. 125
C. 169
D. 189
E. 216

In order to form a Cube the numbers x and y have to be perfect cubes.
Let \(x = a^3\) and \(y = b^3\)
Given \(a^3 - b^3 = 61\)
\((a-b) (a^2+ ab + b^2) = 61 \)
The above equation is a product of two integers and as 61 is a prime number we can conclude
\((a-b) = 1 \) i.e. \(a = b+1 \)
and \((a^2+ ab + b^2) = 61 \)
Substituting a = b+1,
\((b+1)^2 + (b+1)*b + b^2 = 61\)
\(b^2 + 2b + 1 + b^2 + b + b^2 = 61\)
\(3b^2 + 3b + 1 = 61\)
\(b^2 + b - 20 = 0\)
=> b = 4 or -5
Because b should be positive b = 4 and a = 5
\(x= a^3 = 125 \) and \(y = b^3 = 64 \)

Therefore, x+y = 125+64 = 189

Answer: D
Alternatively, substituting the answer choices is quicker.
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