A question that I came across..

I was recently helping a friend out for her queries about how the GMAT exam is like and while looking for sample essays for the AWA part of the test,I stumbled upon a site where I came across a question which I would like to share with you guys.Watch my approach while solving questions of this sort and realized that you do not really need to 'solve' the question for actually getting the answer.The question is-

If x is a positive integer and z is a non-negative integer such that (2,066)^z is a divisor of 3,176,793, what is the value of z^x - x^z?

1)-81

2)-1

3)1

4)0

5)It cannot be determined

How would you go about solving this?

First of all if a question like this appears in the CAT the first thing you have to notice is its purpose.Why do you think this question is there?To test your calculation skills?Not at all.

Calculations of this sort are always easy when they are done on a calculator and the CAT setters wont test you on just your calculation skills.

The point of this question is to check your presence of mind.

If you notice there is no specific value of x which has been given.This should make a thing clear to you.The value of x makes no difference to the answer.But how can that be?After all x occurs two times in critical places in the equation the answer to which is required.

That can occur only in one situation.Did not get it still?

This can only happen when the value of z is one number.What could that number be?Come on it should be easy now.If you still do not know what that value should be scroll down

Yeah if you guessed that value to be 0 you are correct.z has to be 0.The purpose of this question is not to make you find the HCF of 2066 and 3,176,793.Because frankly it could make a fool out of you.The HCF of the numbers if you actually calculate it will turn out to be 1!The numbers are actually coprime.

And that is what makes this question beautiful.

Now when you realize that the value of z is 0 the question becomes very easy to solve.Let x be any number.

We have to calculate z^x - x^z

^{which simplifies to 0x - x^0}

^{which pretty simply is 0-1 i.e. -1.So the answer is option 2.}