A company has ordered 84 sofas, 180 tables, and 264 chairs. It has dep

Hi vmahi77,

This is a "layered" DS question and far more involved than most DS questions that you'll see on Test Day. The 'key' to solving it is to figure out how many departments COULD exist and consider the resulting 'math' for each possibility.

We're told that a company has departments in 4 cities, so we know that there are AT LEAST 4 departments (although a city could have MORE than 1 department within it, so we have to consider that). We're told that each department received the SAME set of furniture, so we can use those furniture numbers to figure out the possible number of departments.

The fastest way to do that is to use prime factorization on the number of sofas, tables and chairs:

84 sofas = (2)(2)(3)(7)

180 tables = (2)(2)(3)(3)(5)

264 chairs = (2)(2)(2)(3)(11)

To figure out the number of departments that COULD exist, we have to pull out the common prime factors. Among the 3 sets, those are....

(2)(2)(3)

We know that there are at least 4 departments (2x2), but there could also be 6 departments (2x3) or 12 departments (2x2x3). The question asks how many departments there are, so the answer will be one of those 3 numbers. The issue is whether more than one possibility exists with each of the given Facts....

Fact 1: One department refused the tables, and those tables were evenly distributed among the rest.

With 4 departments, there would be 45 tables each. If 1 refused its 45 tables, then they would be evenly distributed among the other 3 departments. Thus, 4 is a possible answer.

With 6 departments, there would be 30 tables each. If 1 refused its 30 tables, then they would be evenly distributed among the other 5 departments. Thus, 6 is a possible answer.

With 12 departments, there would be 15 tables each. If 1 refused its 15 tables, then they would NOT be evenly distributed among the other 11 departments. Thus, 12 is NOT possible answer.
Fact 1 has 2 potential answers; Fact 1 is INSUFFICIENT

Fact 2: One department refused the chairs, and those chairs were evenly distributed among the rest.

With 4 departments, there would be 66 chairs each. If 1 refused its 66 chairs, then they would be evenly distributed among the other 3 departments. Thus, 4 is a possible answer.

With 6 departments, there would be 44 chairs each. If 1 refused its 44 chairs, then they would NOT be evenly distributed among the other 5 departments. Thus, 6 is NOT a possible answer.

With 12 departments, there would be 22 chairs each. If 1 refused its 22 chairs, then they would be evenly distributed among the other 11 departments. Thus, 12 is a possible answer.
Fact 2 has 2 potential answers; Fact 2 is INSUFFICIENT

Combined, we know...
There are 4 or 6 possible departments.
There are 4 or 12 possible departments.

Thus, there MUST be 4 departments.
Combined, SUFFICIENT

Final Answer:

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