Hi AbeNeedsAnswers,

We're told that the symbol # denotes one of the four arithmetic operations: addition, subtraction, multiplication, or division and that 6#3 <= 3. We're asked which of the three Roman Numerals must be true.

To start, let's do a bit of analysis on the inequality that we're given (and determine which math operation COULD fit this):
6#3 <= 3

6 + 3 = 9 which is NOT <= 3
6 - 3 = 3 which IS <= 3
(6)(3) = 18 which is NOT <= 3
6/3 = 2 which IS <=3

Thus, the symbol # must be either subtraction or division (although we don't know which one for sure).

I. 2#2 = 0

We've proven that the symbol is either subtraction or division, so let's see if either of those operations 'fits' this information...
2 - 2 = 0
2/2 = 1
Thus, if the symbol is subtraction, then Roman Numeral 1 IS true. However, if the symbol is division, then Roman Numeral 1 is NOT true. There's no way to know which symbol is involved though, so Roman Numeral 1 is not necessarily true.
Eliminate Answers A, D and E.

II. 2#2 = 1

With the work that we've done in Roman Numeral 1 (above), we already know that the outcome of 2#2 could be 0 or 1. This inconsistency also provides that Roman Numeral 2 is not necessarily true.
Eliminate Answer B.

There's only one answer remaining....

III. 4#2 = 2

You can still prove that Roman Numeral 3 is always true...
4 - 2 = 2
4/2 = 2
Both outcomes 'fit' the information in Fact 3, so regardless of what operation is represented, the statement IS true.

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

AbeNeedsAnswers wrote:The symbol # denotes one of the four arithmetic operations: addition, subtraction, multiplication, or division. If 6#3 <= 3, which of the following must be true?

I. 2#2 = 0
II. 2#2 = 1
III. 4#2 = 2

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III

First we need to determine which operation # represents. Since we are given that 6#3 ≤ 3, we can see that # can't be addition or multiplication. However, it can be either subtraction or division. It can be subtraction since 6 - 3 = 3 and it can also be division since 6/3 = 2.

Now let's analyze the given Roman numeral choices.

I. 2#2 = 0

If # is subtraction, then 2#2 = 2 - 2 = 0. However, if # is division, then 2#2 = 2/2 = 1.

We see that I might not be true.

II. 2#2 = 1

If # is division, then 2#2 = 2/2 = 1. However, if # is subtraction, then 2#2 = 2 - 2 = 0.

We see that II might not be true.

III. 4#2 = 2

If # is subtraction, then 4#2 = 4 - 2 = 2. If # is division, then 4#2 = 4/2 = 2.

We see that III is true regardless of whether # is subtraction or division.

Answer: C