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14 Aug 2026, 10:34
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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41%
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Consider the equation:
1/x + 1/y = 1/z, where x, y, and z are positive integers.
Which of the following could be the value of x + y + z?
I. 4
II. 10
III. 20
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
1/x + 1/y = 1/z, where x, y, and z are positive integers.
Which of the following could be the value of x + y + z?
I. 4
II. 10
III. 20
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
14 Aug 2026, 12:44
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Bunuel
At first, it may seem like you'll need to guess and check with a lot of numbers against all three Roman numerals. In this situation, make your job easier by using reasoning and pattern recognition to avoid wasting time trying random combinations.
The first value (4) is the the smallest, so it should be easiest to determine. Let's start there.
I. 4
For x + y + z to equal 4, all the integers must be very small. Test the smallest positive integer, 1. If x = 1 or y = 1, then 1/x + 1/y will be over 1. But since z is also a positive integer, the largest value possible for 1/z is 1 (when z = 1). So neither x nor y can be 1.
If x = 2 and y = 2, then z must be 1:
1/2 + 1/2 = 1/1
But this means that x + y + z = 5. This is the smallest value you can get for x + y + z, so you can eliminate statement I (and choices A and D).
Notice, however, that when x and y were both 2, z was equal to half of 2, yielding 2 + 2 + 1 = 5. Could this be a pattern?
II. 10
In fact, it is a pattern. If you simply double all the values, you have 4 + 4 + 2 = 10:
1/4 + 1/4 = 1/2
Statement II "works", and you can eliminate choice C.
III. 20
To get to x + y + z = 20, you can simply double all the values again:
1/8 + 1/8 = 1/4
Statement III also works. The answer is E.
When you're testing numbers, use reasoning and look for patterns to avoid having to try multiple sets of random numbers.
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