Bunuel wrote:
For integers x and y, if 91x = 8y, which of the following must be true?
I. y > x
II. y/7 is an integer
III. The cube root of x is an integer
A) I only
B) II only
C) III only
D) I and II
E) II and III
Without actually solving the equation algebraically, we can see that x could be 8 and y could be 91, since 91(8) = 8(91). However, this is not the only possibility. We see that x could be 0 and y could be 0, since 91(0) = 8(0), or x could be -8 and y could be -91, since 91(-8) = 8(-91). In any event, we see that x is a multiple of 8 (including 0 and the negative multiples) and y is a multiple of 91 (including 0 and the negative multiples).
Let’s analyze each Roman numeral:
I. y > x
Since both x and y could be 0, y is not necessarily greater than x.
Roman numeral I is not true.
II. y/7 is an integer
Since y is a multiple of 91, y/7 is an integer.
Roman numeral II must be true.
III. The cube root of x is an integer
We’ve mentioned that x is a multiple of 8. If x = 8, then the cube root of x is an integer. However, if x = 16, then the cube root of x is not an integer.
Roman numeral III is not true.
Answer: B
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