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27 May 2019, 06:15
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Difficulty:
95%
(hard)
Question Stats:
29% (01:59) correct
71%
(02:05)
wrong
based on 112
sessions
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How many rectangles with area 14,400 have integer lengths and widths?
A) 31
B) 32
C) 33
D) 62
E) 63
A) 31
B) 32
C) 33
D) 62
E) 63
Archit3110
Major Poster
Updated on: 27 May 2019, 08:42
Originally posted by Archit3110 on 27 May 2019, 06:22.
Last edited by Archit3110 on 27 May 2019, 08:42, edited 1 time in total.
Last edited by Archit3110 on 27 May 2019, 08:42, edited 1 time in total.
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GMATPrepNow
im not sure of the solution
we factorize 14400= 2^6*5^2*3^2
7*3*3 = 63
since we need individual l & w sides as integers so 63/2; 32.5
at max we can have 31 rectangles pairing of 31 possible integers is possible and 32 by 32 squares
answer will be either of A & B
27 May 2019, 07:39
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As a question design note, I'm not sure it makes sense to pose a question this way - what makes one rectangle different from another? If I draw one 14,400 by 1 rectangle in the coordinate plane, say, with its bottom-left vertex at (0,0), isn't that a different rectangle from one with the same dimensions with its bottom-left vertex at (100000, 1000000)?
But if we take the question to mean: how many distinct integer dimensions L by W of a rectangle are possible, if the rectangle's area is 14,400, and L > W (length must be greater than or equal to width)? then we can prime factorize 14,400:
14,400 = 144 * 100 = 12^2 * 10^2 = (2^2 * 3)^2 * (2*5)^2 = 2^6 * 3^2 * 5^2
Now adding one to each exponent and multiplying, we find the total number of divisors of 14,400; 14,400 has (7)(3)(3) = 63 divisors. If we were to list all of these divisors, they would all be in pairs that produce 14,400 as a product:
14,400 times 1
7200 times 2
etc
with one exception: because 14,400 is a perfect square, 120^2, we have one unpaired divisor. So we have 62 divisors in pairs, and since we presumably don't want to consider a 14,400 by 1 rectangle to be different from a 1 by 14,400 rectangle, we'll get 31 distinct rectangles from those divisors, and then we lastly have the 120 by 120 square (which is also a rectangle, since a rectangle is simply a quadrilateral with four right angles). So we have 32 distinct rectangles in total with integer dimensions, as I've interpreted the question.
But if we take the question to mean: how many distinct integer dimensions L by W of a rectangle are possible, if the rectangle's area is 14,400, and L > W (length must be greater than or equal to width)? then we can prime factorize 14,400:
14,400 = 144 * 100 = 12^2 * 10^2 = (2^2 * 3)^2 * (2*5)^2 = 2^6 * 3^2 * 5^2
Now adding one to each exponent and multiplying, we find the total number of divisors of 14,400; 14,400 has (7)(3)(3) = 63 divisors. If we were to list all of these divisors, they would all be in pairs that produce 14,400 as a product:
14,400 times 1
7200 times 2
etc
with one exception: because 14,400 is a perfect square, 120^2, we have one unpaired divisor. So we have 62 divisors in pairs, and since we presumably don't want to consider a 14,400 by 1 rectangle to be different from a 1 by 14,400 rectangle, we'll get 31 distinct rectangles from those divisors, and then we lastly have the 120 by 120 square (which is also a rectangle, since a rectangle is simply a quadrilateral with four right angles). So we have 32 distinct rectangles in total with integer dimensions, as I've interpreted the question.
