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.}

Two revelations..

I have wanted to talk about two things since the day I started this blog.Two things which become very helpful when it comes to solving questions like "Find the last two digits of 16^54(Lets say)".This question can have many forms to it like finding the last 2 digits of odd numbers raised to some power,finding the last 2 digits of a number which is a product of an odd number and an even number raised to a power and also finding the last 2 digits of a number raised to a power which is itself a power of some number(Say the last 2 digits of 82^82^82).

Anyway the 2 fundae are-

1)Any number which ends with 76 when raised to any power will always end with 76!

Go on try it.76^2 is 5776 like we found in the last post.176^3 is 959512576.I can go on but I guess you should believe me by now.

2)A number ending with 24 when raised to a power will always end with either 76 or 24.It will end with 76 when it is raised to an even power.And when it is raised to an odd power it will end with 24.

Let us check this out.24^2 is 576.24^3 is 13824.Again I can go on.

The application?The amazing thing is that every even number can be broken into a power of 2 multiplied by some constant.And every 10th power of 2 will end with 24!And the square of 24 will always end in 76!Which makes questions of the sort I talked about in the beginning very easy to solve.Lets solve the question I spoke of in the beginning to understand it.To find the last 2 digits of 16^54 I will do the following-

a)Break 16 into powers of 2 by writing it as 2^4 i.e. 16^54 become 2^216.

b)Break 2^216 as 2^210*2^6.

c)Now 2^10 is a number ending with 24.It means that I can write 2^210 as (....24)^21.

d)As made clear by the second funda,this is a number ending with 24 and raised to an odd power.Therefore the last 2 digits here would be 24.

e)The last 2 digits of 2^6 are 64(The only 2 digits).

f)Therefore the last 2 digits of the number in the question will be the last 2 digits of the multiplication of 64 and 24 which gives us 1536.

g)Hence the answer is found to be 36.

These 2 things turn even more useful when combined with another funda I will talk about later wherein we look at finding the last two digits of an odd number raised to any power.

Subsequently we will also look at ways to solve the questions of the 3rd type wherein the power a number to raised to is a power in itself.

But till the next time,I hope you liked this.

A Facebook Group to carry this forward!

Just to inform everyone,me and a couple of my friends have created a Facebook group just to carry this initiative forward.So all of you guys who are on Facebook can join the group here-

http://www.facebook.com/group.php?gid=120181764668387

The group is called Top CAT and we would be very regular with it.Join it and have fun preparing.