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
_________________

I solved it this way. I'm sure it's not the perfect approach.

Look the two statements that are given - there's no way we can answer the question using only one of these statements. Hence, A, B and D cannot be the answer.

Now, consider both the statements together. What we can say is that if there's one less department that the number actually present, the chairs (264) and tables (180) can be equally divided. Now, if you look at numbers which divide both 264 and 180, there are a lot; I assumed 6 and 12 as a small sample set.

Hence, 7 and 13 can be the number of departments across the four cities. Now these can be distributed in any manner (We're not really concerned with that). Because more than one value can fit the situation, we cannot zone in on one absolute value. Hence, I marked 'E' as the answer.

Is my approach right? Anyone?

How is it okay to assume that the minimum number of departments is 4?

I could have the same department spread across multiple cities. Say, my sales team is across City A and city B and that I actually split the furniture across the two divisions of the same department in ways not mentioned in this question premise.

Hi rajarams,

You bring up an interesting point. To that end, if this question appeared on the Official GMAT, then the writers probably would have stated "there is at least one department in each of 4 cities" as a means to clarify the issue.

That having been said, your way of interpreting the question actually ignores a vital piece of information that you're given to work with. If a department could have been spread across two (or more) cities, then the number of cities would not matter. The result of that idea should have prompted you to rethink the way that you were interpreting the question.

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

Yes. To be frank, I missed the minimum number of departments point when I marked E as the answer. But then when I looked up the solution, I was not sold on this. For the question, in its original form, I still think E is the right answer. I guess the question premise did not place a lot of emphasis on the cities connection so that it gets ignored by most test takers. If it were made clearer as you had suggested, it is no more as much of a trap question as the original one is.

This question has a flaw in itself in not mentioning the minimum number of departments each city can have.
Answer would be E since there's no information about the min no of departments. If this is the case, it can be 2,3,4,6,12( HCF). Answer would be C if the information has been given about the min no of departments in each city. In this case it will be 4.

Although the question might miss a few information, this is a very good one testing the concept of HCF.

The common multipliers that are not less than 4 are 4,6,12
4 departments: 21, 45, 66
6 departments: 14, 30, 44
12 departments: 7, 15, 22

statement one gets rid of 12 departments option as 15 (refused tables)/ 11(remaining dept) is not an integer
statement two gets rid of 6 departments options as 45(refused sofas)/5(remaining dept) is not an integer

hence 4 departments is the answer ie c

From the stem, it can be inferred the company has at least 4 departments. Lets find the common multiples of the furniture.
Sofas: 84 = 4 * 21 = 4 * 3 * 7
Tables: 180 = 4 * 45 = 4 * 3* 3 * 5
Chairs: 264 = 4* 66 = 4 * 3 * 2* 11

Common multiples are: 4 and 12, so no of departments has to be either 4 or 12.

1) If the company has 4 departments, it can distribute 45 tables to each department. So, if one department refused the tables, the other 3 department can receive the remaining 45 tables. Again, if the company has 12 departments, the other 11 departments cannot divide the allotted 15 tables per department. So, the company has 4 departments. Sufficient.

2) If the co. has 4 departments, each will get 66 chairs, which can be divisible among the other 3 departments if one department refuses the chairs. If the co has 12 departments, the allotted chairs is 22, which can also be divisible among other 11 departments. Not sufficient.

A is the answer.

Hi minustark,

Most of your work here is perfect, BUT you missed one option - there could be 6 total departments (not just 4 or 12) and that impacts the answer to the question.

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

Therefore, the number of departments must be a common factor of 84 180, 264.

84=2*2*3*7
180=2*2*3*3*5
264=2*2*3*2*11

Therefore, the common prime factors are 2, 2 and 3.
Common factors are a combination of these, 2*2*3, 2*3, 2*2.
Hence, the common factors are 12, 6 and 4.

-------------------------

(1)
So the number of tables must be divisible by one of the common factors as well as the (common factor-1).

180 is divisible by 12, but not by 11, so 12 is out.
180 is divisible by 6 as well as by 5.
180 is divisible by 4 as well as by 3.

So the possible number of departments are 6 and 4.

Hence (1) is insufficient.

-------------------------

(2)
Following the same procedure as (1),

264 is divisible by 12 as well as by 11.
264 is divisible by 6, but not by 5, so 6 is out.
264 is divisible by 4 as well as by 3.

So the possible number of departments are 12 and 4.

Hence (2) is insufficient.

-------------------------

(1,2)
Based on (1), the possible number of departments are 6 and 4.
From (2), the possible number of departments are 12 and 4.

The common answer is 4.
Hence, the number of departments possible is 4.

Option C

the number of department can be 3 and 2 as well in such case S, T, C will be 28,60,88 and 42,90, 132. In such case the answer will be inconclusive which is E. Please correct me if I am wrong.

Hi thakurarun85,

The prompt tells us that the company has departments in 4 cities, so there cannot be just 2 or 3 departments (there has to be at least 4).

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