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If the lengths of perpendicular sides of a park that is in the shape of a right-angled triangle park are integers and

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by Gmat_mission » Thu May 21, 2020 1:17 am

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If the lengths of perpendicular sides of a park that is in the shape of a right-angled triangle park are integers and the area of the triangular park is 630 square units, which of the following can be the greatest number that divides the lengths of both the perpendicular sides?

I. 6
II. 10
III. 21


A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

[spoiler]OA=A[/spoiler]

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Re: If the lengths of perpendicular sides of a park that is in the shape of a right-angled triangle park are integers an

by Scott@TargetTestPrep » Sun May 31, 2020 9:52 am
Gmat_mission wrote: ↑
Thu May 21, 2020 1:17 am
 If the lengths of perpendicular sides of a park that is in the shape of a right-angled triangle park are integers and the area of the triangular park is 630 square units, which of the following can be the greatest number that divides the lengths of both the perpendicular sides?

I. 6
II. 10
III. 21


A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

[spoiler]OA=A[/spoiler]

Solution:

If the two perpendicular sides of the right triangle are a and b, we have:

½ x a x b = 630

a x b = 1260

Now let’s prime factorize 1260:

1260 = 126 x 10 = 6 x 21 x 2 x 5 = 2 x 3 x 3 x 7 x 2 x 5 = 2^2 x 3^2 x 5 x 7.

We see that 6 can be the greatest number that divides the lengths of both sides if, for example, a = 2 x 3 x 5 and b = 2 x 3 x 7. This is possible because 1260 has two factors of 2 and two factors of 3. Therefore, we see that neither 10 nor 21 can be the greatest number that divides the lengths of both the perpendicular sides since there is only one factor of 5 and one factor of 7 in 1260.

Answer: A

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