In the figure above, all angles are right angles. If the len

[BTGmoderatorDC]

In the figure above, all angles are right angles. If the lengths a, b, u, and v are integers, what is the perimeter of the six-sided figure?

(1) ab = 77

(2) u/v = 3/5

OA A

Source: Manhattan Prep

[Ian Stewart]

The perimeter of the figure is identical to the perimeter of a rectangle with sides a and b (if you just draw the two lines in the top right corner that you'd need to draw to make that rectangle, you'll see that you're adding a line of length v to the top side, and a line of length u to the righthand side). So we don't care much about u and v here.

If Statement 1 is true, then the only integer values a and b can have are (in some order) 7 and 11, or 1 and 77. But they can't be 1 and 77, because if, say, a = 1, then we'd have that u < 1, and u also needs to be an integer. Similarly if b=1, then v < 1, and v needs to be an integer. So a and b are 7 and 11 (or 11 and 7) and the perimeter must be 2*7 + 2*11 = 36.