In an auditorium, 360 chairs are to be set up in a rectangular arrange

Question

In an auditorium, 360 chairs are to be set up in a rectangular arrangement with x rows of exactly y chairs each. If the only other restriction is that 10 < x < 25, how many different rectangular arrangements are possible?

A. Four
B. Five
C. Six
D. Eight
E. Nine

PS66602.01

A. Four
B. Five
C. Six
D. Eight
E. Nine

PS66602.01

ZoltanBP · Accepted Answer

x and y are positive integers such that 360 = xy.

The rephrased question: How many factors does 360 have between 10 and 25, exclusive?

Since  360=2^3\cdot 3^2\cdot 5 , we can quickly list the suitable factors.

3\cdot5=15 
 2^2\cdot5=20 
 2\cdot3^2=18 
 2^2\cdot3=12 
 2^3\cdot3=24

Answer: B

ScottTargetTestPrep · Answer

Since you know there are (3+1)(2+1)(1+1) = 24 possible values of x, I understand you already factored 360 = 2^3 * 3^2 * 5.

First of all, you definitely don't need to consider all 24 possibilities for x since we are only interested in the values of x that satisfy 10 < x < 25. In order to make sure we are not missing any factor pairs, we will find all values of x in the interval 10 < x < 25 that are a factor of 360.

We already have the prime factorization 360 = 2^3 * 3^2 * 5; therefore we can immediately eliminate the primes in the interval 10 < x < 25 which do not appear in the prime factorization of 360. Namely, we eliminate the values x = 11, 13, 17, 19 and 23. Next, we can eliminate any multiples of 7 and 11 as well (since 360 cannot be divisible by any multiple of 7 or 11). Thus, the values x = 14, x = 21 and x = 22 are also eliminated. We are left with x = 12, 15, 16, 18, 20, 24. At this point, we can simply check these six values and determine that 360 is divisible by all except 16 (too few factors of 2 in 360 to be divisible by 16). This is the fastest way I can come up with which will guarantee that you won't miss any factor pairs.

Archit3110 · Answer

360 will have ; 2^3*3^2*5^1 ; 24 factors
possible integers for array x*y where x has restrictions 10<x<25 ; 12,15,18,20,24 ; 
5 options
IMO B

BrentGMATPrepNow · Answer

From the given information, the TOTAL number of chairs = xy
This means: xy = 360

Since x and y must be POSITIVE INTEGERS, there is a finite number of possibilities. 
To help us list the pairs of values with a product of 360, let's find the prime factorization of 360
360 = (2)(2)(2)(3)(3)(5)

When we consider the fact that 10 < x < 25, the possibilities are:
x = 12 & y = 30
x = 15 & y = 24
x = 18 & y = 20
x = 20 & y = 18
x = 24 & y = 15

There are five such possibilities

Answer: B

Cheers,
Brent

ScottTargetTestPrep · Answer

We need to find integers x and y such that xy = 360 and 10 < x < 25. Since

360 = 12 * 30 = 15 * 24 = 18 * 20 = 20 * 18 = 24 * 15

We see that x can be 12, 15, 18, 20, or 24. Therefore, there are 5 different arrangements.

Answer: B

dabaobao · Answer

ScottTargetTestPrep  This question is simple but took a bit of time in order to avoid missing any values of x. While I know how to solve it instantly by taking factor pairs, it did take me 3 min for me to double check to make sure I didn't miss any value of x in that range. Is there a quick to find all values of x that satisfy the condition 10 < x < 25 since we can have (3+1)(2+1)(1+1) = 24 values of x? Thanks!

Abhishek009 · Answer

Possible values of (X , Y) are -

1. ( 12 , 30)
2. ( 15 , 24)
3. ( 18 , 20)
4. ( 20, 18 )
5. (24 , 15 )

Hence, 5 arrangements are possible, correct Answer must be (B)

fireagablast · Answer

X rows by Y chairs = X*Y
X*Y = 360

Prime factors of 360 = 2*2*2*3*3*5
X values allowed 10<x<25 = 11 through 24
values of X that you can build with the prime factors of 360
12, 15, 18, 20, 24

5 values of x

Cellchat · Answer

We need to find integers x and y such that xy = 360 and 10 < x < 25.

360 =  2^3 3^2 5

Any value of x that contains a prime factor that is not 2, 3, or 5 will not satisfy the condition of xy = 360. Any value of x that contains a higher order prime than those in 360 will also not satisfy the condition of xy = 360. For example, 16 =  2^4  and hence is not a value of x. 14 = 2 x 7 and hence is not a value of x.

x =  11 ,  12 ,  13 ,  14 ,  15 ,  16 ,  17 ,  18 ,  19 ,  20 ,  21 ,  22 ,  23 ,  24

Number of rectangular arrangements = 5 --> (B)

Hope it helps.

ArindamLucky · Answer

Values of x which divide 360-- > 
12,15,18,20,24

Posted from my mobile device

AyeeshaJ · Answer

Hey  ScottTargetTestPrep  &  Bunuel ,

I have learned on GMAT Club that the number of ways to write a Number as a product of two factors is (Total Factors/2).

360 has 24 factors, so it should be possible to write 360 as a product of two factors in 12 different combinations.

However, when I tried to list them down, the number of such combinations was much greater. Where am I going wrong?

This is a conceptual gap that is challenging my approach to many questions. Please answer!

MathRevolution · Answer

Total chairs: 360

Total rows: 'x' with 'y' chairs each. Hence, x * y = 360

=> 10 < x < 25. Therefore possible 'x' are from 11 to 24 and possible 'x' which will be factor of 360 are: 12, 15, 18, 20, 24.

Answer B

Basshead · Answer

The auditorium has 360 chairs.

Prime factorization of  360 = 2^3* 3^2 * 5

The chairs are set up in an arrangement of x rows of exactly y chairs. 10 < x < 25

For there to be x rows of y chairs, the factors of x must be within 2^3* 3^2 * 5. In other words, there can't be a number such as a 5, or 7 because there would be left over chairs.

We're told 10 < x < 25

11 ,  12 ,  13 ,  14 ,  15 ,  16 ,  17 ,  18 ,  19 ,  20 ,  21 ,  22 ,  23 ,  24 ,  25

Only five numbers in the given range have a prime factorization that fits. Answer is B.

Shrey08 · Answer

Do you agree that "360 will have; 2^3*3^2*5^1; 24 factors" isn't really helpful, given the restrictions for x?

Or did I miss your point?

ravi1522 · Answer

As there is total 360 chairs 
so X*Y =360 
as y is integer then 360/x has be integer so what all value of x is factor of 360 
Hence x will be 12,15,18,20,24 hence 5 rectangular patterns can be made.

Hit Kudos, if it is useful.

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In an auditorium, 360 chairs are to be set up in a rectangular arrange

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