## An Often Overlooked GRE Strategy

Sometimes, on the New GRE math, we are tempted to try to do all the work in our head. Depending on the problem – and your ability to do mental math – working a problem out in your head can be time efficient. However, many of the problems on the GRE are meant to trick you, or be too convoluted for most people to do in their heads.

Instead – especially for complicated problems – we have to rely on another strategy. Sure, it’s not as glamorous as doing extreme number crunching in your head à la Rain Man, but writing down steps and important numbers is a very useful strategy. And, by important numbers, I mean those numbers that are part of the solution to the problem, but are not the actual solution themselves.

**Round 1 – Meet the Symbol (And Get Your Pen(cil) Ready)**

Now that I’ve prefaced the strategy of writing down the problem and important numbers, try to do the following problem in your head. If you can’t crack it – or if your head starts spinning in circles, and you eye the closest exit – then try writing down your steps to see if that helps you arrive at the answer.

1. P* is defined as the number of positive even integers less than P, if P is odd. If P is even, P* is defined as the number of prime integers less than P. What is (5* + 10*)*?

(A) 3*

(B) 4*

(C) 7*

(D) 10*

(E) 11*

If you got answer (C), good job (and if you did this in your head – very good job)! For most, though, this is a tricky problem with a lot of steps, so it’s best to write down a few important numbers along the way. For instance, you might want to write down 5* = 2, and 10* = 4. (2 + 4)* = 6*, which equals 3. The only answer choice that yields 3 is 7*. Had you, of course, tried to balance all of that in your head, you may have picked the wrong answer, or simply gotten another answer altogether.

By the way, this type of problem is called a symbol, or “funky symbol” problem by some. Here the symbol is *. The key is remembering that the symbol has no mathematical function beyond what the problem specifies.

**Round 2 – An Even More Frightening Symbol**

Let’s try another symbol problem. Again, you can attempt to do it in your head, but I encourage you to try writing down a few of your steps. Let’s see who can get the correct answer (with explanation) first!

2. ^m^ is equal to the digits in positive integer m in reverse order, discounting the zeroes (e.g. ^41^ = 14 but ^3500^ = 53). Which of the following must be true? Select all that apply.

- ^m^ < ^m+1^
- m = ^(^m^)^
- ^1000m^ = ^m^
- (^m^)(^m^) > ^m^

The chances of answering this diabolical Multiple Answer Question (MAQ) correctly without writing anything down are very slim. A great strategy is to plug in numbers for each problem, seeing if you can disprove A – D. Why disprove? The question says which of the following MUST be true. As soon as you can find an instance where the equation is not correct, you can discount that letter as one of the answer choices.

With A we could plug in 99 for m. This would give us 99 on the left-hand side and ^100^ = 1, on the right hand side.

With B, we can plug in 100, which would give us m = 100. ^(^m^)^ = 1. Therefore, the two sides are not equal.

With C, no matter which number you plug in for m, both sides will always be equal. By multiplying m by 1000, you are simply omitting those zeroes when you reverse the digits. For instance, if m = 35, then ^35000^ = ^35^. Both sides equal 53. Therefore, (C) is an answer.

For D, we can disprove it by plugging in 10, or, for that matter, any number beginning with 1 and ending with any number of zeroes.

*GRE Strategy Takeaways*

**Unless you are short on time, try to write down important steps in a problem.**

**For symbol problems, read the problem carefully to make sure you understand the function of the symbol.**

**For MUST questions, the best strategy is to disprove, not confirm, answers.**

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Why is 10*=4? Aren’t the positive prime integers less than 10 1,2,3,5 and 7? Then it should equal 5.

Thanks for your help.

Hi Betsy,

This is certainly a tricky question! The prime integers less than 10 are 2,3,5 and 7. 1 is NOT a prime number, so it’s not counted here. This website provides some useful facts about 1 and why it’s not a prime number 🙂

P* is defined as the number of positive even integers less than P* if P is odd, and if P is even, P* is defined as the number of prime integers less than P*. What is (5* + 10*)*?

This question is likely to be wrong:

this should be like this:

P* is defined as the number of positive even integers less than P if P is odd, and if P is even, P* is defined as the number of prime integers less than P. What is (5* + 10*)*?

Hi BD,

Good catch! That is definitely confusing since P* is not a number; P is a number. I’ll make the corrections right away :).

The explanation really helped, Thanks Chris .

I’m having hard time to understand this question….P* is defined as the number of positive even integers less than P* if P is odd, and if P is even, P* is defined as the number of prime integers less than P*. What is (5* + 10*)*?

can i pls know what exactly this question is asking.

Let’s say p is 7, an odd number. As defined by the operation p*, we wil interpret at odd number different from the way we would an even integer. For 7*, we want to know wow many positive even integers are less than 7. We get 2, 4, 6 gives us a total of 3 even integers. Therefore 7* = 3.

Conversely, if p is an even integer, say 8, then how many prime numbers are there less than p? 2, 3, 5, 7. So 8* = 4.

Apply this to the question above we get 5* = 2. 10* = 4. (2 + 4)* = 3, which is the same as 7* = 3. Answer (C).

I hope that makes it easier to understand the question. Let me know!

I was having a hard time understanding the original question as well. But it’s much more clear after that explanation, thank you.

Great! I am happy that made more sense!