Each of the math questions below is directly inspired by a question in the on-line Revised GRE test. I’ve provided an easier version of the question (#1) and a more difficult version of the question (#2).
My recommendation is to try the easier version first. Then, if you answer it correctly, click on the link, and take a stab at the actual Revised GRE question.
If you are able to answer that question correctly, then as prize – you get a fiendishly difficult question (#3). Okay, maybe that’s not a prize – but it is great practice for those aiming for the 90% on quant.
The good news is I have explanations. For the Revised GRE question, I have recorded an explanation video you can watch. Finally, it is a good idea to try the easy question before the medium one, and the medium question before the difficult one.
1. Difficulty: Easy
If , where n is a non-negative integer, what is the greatest value of ?
Explanation: Don’t think big – think small. That is the smaller n becomes the greater ½^n becomes. So what is the smallest value? You may be tempted to say 1, which would give us ½. But remember n = 0, because . Therefore Answer: B.
The “hidden zero,” as I like to call it, is a classic GRE math trick. So always keep your eyes open, especially when you see “non-negative integer,” which includes zero.
2. Difficulty: Medium-Difficult
(5/4^-n) < 16^-1
What is the least integer value of n?
The best place to start here is by getting rid of the unseemly negative signs and translating the equation as follows:
(4/5)^n < 1/16
A good little trick to learn using 4/5 taken to some power is that (4/5)^3 = 64/125, which is slightly—but only slightly—greater than ½. Therefore, we can translate (4/5)^3 to ½.
(1/2)^4 = 1/16
That would make (4/5)^12 a tad larger than 1/16. To make it less than 1/16 we would multiply by the final 4/5, giving us n = 13.
3. Difficulty: Hard
The equation is true for how many unique integer values of n, where n is a prime number?
- None of the above
This problem can be difficult, indeed downright inscrutable, unless you take your time and process one piece of information at a time. Once you understand what the problem is saying, you should be able to solve the question relatively quickly.
The most important piece of info is n is a prime number. So do not start by plugging in zero or one. Neither is a prime. The lowest prime is 2. When we plug in ‘2’ we get:
This is clearly true. Thus we have one instance.
As soon as we plug in other prime numbers a pattern emerges.
is always a negative number if n is odd. Because all of the primes greater than 2 are odd, the number in the middle will always be negative:
Because in each case n is a positive number we can never have the middle of the dual inequality be positive, if n is an odd prime.
Thus the only instance in which the inequality holds true is if we plug in ‘2’, the answer is (D).
If you got that right – congratulate yourself. It’s a toughie.