If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of thefollowing statements must be true?I. There is only one other rectangle that has integral dimensions and the same area as the given rectangleII. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangleIII. The area of the given rectangle is 3599 square inches.a)I onlyb)II onlyc)III onlyd)I and II onlye)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

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If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of the
following statements must be true?
I. There is only one other rectangle that has integral dimensions and the same area as the given rectangle
II. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangle
III. The area of the given rectangle is 3599 square inches.

a)
I only
b)
II only
c)
III only
d)
I and II only
e)
None of the above

Correct answer is option 'A'. Can you explain this answer?

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If the dimensions of a rectangle (in inches) is equal to a prime numbe...

Given:

Length of the rectangle (inches) = {59, 61}
Breadth of the rectangle (inches) = {59, 61}
Note that we’re not given that Length and Breadth are distinct integers.

So, it would be wrong to assume so.

To Find: Which of the 3 given statements must be true?

Approach:

We’ll evaluate the 3 statements one by one. They will qualify as must be true statements only if no exception is found to those statements.

Working out:

Evaluating Statement I
- There is only one other rectangle that has integral dimensions and the same area as the given rectangle
- This statement talks about the area of the given rectangle. So, let us first see what are the different possible values for the area of the given rectangle. We find that 3 values of area are possible:
59*59
59*61
61*61
Let’s evaluate the first possibility, where the area of the given rectangle is 59*59

In how many ways can 59*59 be expressed as a product of 2 integers?

Way 1 - (59)*(59)
Way 2 - (1)*(59*59)

This means, only 2 rectangles are possible whose area is 59*59:
- A rectangle each of whose side is 59 inches
  - This is the given rectangle
- A rectangle whose one side is 1 inch and the other is 59*59 inches
Thus, apart from the given rectangle, there is only one other rectangle that can have the same area as the given rectangle
We can do a similar analysis for the other 2 possible values of the area (59*61 and 61*61). We’ll get the same result – that there is only one other rectangle that can have the same area as the given rectangle
- The reason why the other 2 values will also yield the same result is that these 2 values of the area too are a product of prime numbers, like the first possibility was. So, the analysis and the conclusion will be similar.
Thus, Statement 1 holds true for each of the 3 possible values of the area of the given rectangle. Therefore, this statement is a must be true statement.
Evaluating Statement II
There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangle
This statement talks about the perimeter of the given rectangle. So, let’s first evaluate what the perimeter of the given rectangle will be. We find that 3 values are possible for the perimeter:
- 2(59+59)
- 2(59+61)
- 2(61+61)
Let’s evaluate the first possible value of the perimeter, 2(59+59) and see
how many different rectangles can have this perimeter
All the following pairs of length and breadth lead to the same perimeter:
- (59 – 1, 59 + 1), that is, (58, 60)
- (59 – 2, 59 + 2), that is, (57, 61)
- (59 – 3, 59 + 3), that is, (56, 62)
- And so on . . .
Thus, we see that there are more than one rectangles that can have the same perimeter as the given rectangle if the perimeter of the given rectangle is 2(59+59)
Since Statement II has proved untrue for one possible value of the Perimeter, it is definitely not a must be true statement.
Note: You may have done your analysis with one of the 2 other possible values of the perimeter. That’s perfectly okay. The line of reasoning will be the same and will lead to the same conclusion as above.

Evaluating Statement III
The area of the given rectangle is 3599 square inches.
As discussed in Statement I, 3 values of Area of the given rectangle are possible:
- 59*59
- 59*61
- 61*61
Therefore, we cannot say definitely what the area of the given rectangle is. We can only say that the area can be 59*59 or 59*61 or 61*61 square inches
So, Statement III is not true.

Looking at the answer choices, we see that the correct answer is Option A

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If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of thefollowing statements must be true?I. There is only one other rectangle that has integral dimensions and the same area as the given rectangleII. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangleIII. The area of the given rectangle is 3599 square inches.a)I onlyb)II onlyc)III onlyd)I and II onlye)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

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If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of thefollowing statements must be true?I. There is only one other rectangle that has integral dimensions and the same area as the given rectangleII. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangleIII. The area of the given rectangle is 3599 square inches.a)I onlyb)II onlyc)III onlyd)I and II onlye)None of the aboveCorrect answer is option 'A'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of thefollowing statements must be true?I. There is only one other rectangle that has integral dimensions and the same area as the given rectangleII. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangleIII. The area of the given rectangle is 3599 square inches.a)I onlyb)II onlyc)III onlyd)I and II onlye)None of the aboveCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the dimensions of a rectangle (in inches) is equal to a prime number that lies between 55 and 65, exclusive, which of thefollowing statements must be true?I. There is only one other rectangle that has integral dimensions and the same area as the given rectangleII. There is only one other rectangle that has integral dimensions and the same perimeter as the given rectangleIII. The area of the given rectangle is 3599 square inches.a)I onlyb)II onlyc)III onlyd)I and II onlye)None of the aboveCorrect answer is option 'A'. Can you explain this answer?.

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