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.

Bonus for the weekend!

I thought of adding another small trick which is pretty helpful as the week comes to an end.This has also been a pretty useful trick but has an underlying assumption.The trick is about calculating square on a number given the square of the preceding or the succeeding number.The funda comes from a formula which most of us would have done as a start to algebra in class 6th.The formula is and I think all of you would remember it-

a^2-b^2=(a-b)(a+b)

For the uninitiated ^ implies the 'raised to the power of ' sign.

Funda-To calculate the square on a number without multiplying it by itself you can use the following technique if you have the square of the preceding number with you.

1)Add the 2 consecutive numbers

2)Add the sum to the square of the preceding number.

3)In this case the (a-b) term becomes 1 and that is what makes the difference of the squares the sum of the numbers.

Let me elucidate with an example.

Example-Let me calculate the square of 76 for you guys.I hope everyone knows the square of 75.If you dont you can learn this trick about calculating squares of numbers ending with 5(Believe me this is justs the tip of the iceberg).The square will always end with the last two digits as 25.The first few digits can be calculated by multiplying the number preceding 5(in this case 7) with the next number(in this case 7+1=8).The multiplication of 7 and 8 will give us 56 which become the first 2 digits of the square giving us the answer as 5625.

Now that we have the square of 75 we follow the steps mentioned

1)Add 75 and 76 to get 151

2)Add 151 to the square of 75 i.e. 5625

3)The sum i.e. 5776 is your answer

The underlying assumption that I mentioned before is that you should be quick enough with your additions and should know the squares of some important numbers atleast.

Hopefully you enjoyed this funda and can come back next week when I really start with some amazing revelations.Until then Happy Solving!

A gentle start..

Let me talk about two pretty basic things today.These things may not have a lot of direct applications but you will find them useful if you begin using them regularly.In my experience,they have helped me a lot in doing my calculations faster and hopefully they will help you guys too.Here it goes-

a)The average of two numbers-

This trick will help you calculate the average of two numbers in a different way from just adding them and dividing them by 2.I will show you the application after I highlight the funda.Lets say we have two numbers a and b.We can find their average by-

1-Subtract the smaller number from the larger one

2-Divide the difference by 2.

3-We get the average by either adding the answer obtained in 2 to the smaller number or subtracting the same from the larger number.

For example-I have two numbers 4 and 8.I subtract 4 from 8 to get 4.I divide 4 by 2 to get 2.And I add 2 to 4 or subtract 2 from 8 to get the average as 6.

Application-Now this seems pretty weird right?I mean its basically increase of a step from the general method.But the application comes in when you look to calculate the average of 2 numbers which are easier to subtract than add.

Case in point-999 and 1999.The difference is 1000.I get 500 when I divide it by 2.And when I add it to 999 or subtract it from 1999 I get the average which is 1499.

Believe me once you start using this frequently you will become so comfortable with it that you will use it subconsciously.

Note-This can only be used to find the average of 2 numbers.

b)Digital Sum

First of all let me explain what digital sum is.It is as the name specifies the sum of all the digits of a number until you have just a single number.

For example-The digital sum of 887 is 5(8+8+7=23;2+3=5)

Now that you know what digital sum is the point I want to make is-

Funda-The sum of the digital sum of 2 numbers is the digital sum of the sum of the same 2 numbers.

Confused?Read it again and understand it.

Example-What is the sum of the digital sums of 23 and 45?

Now either you can calculate the digital sums of 23 and 45 separately(5 and 9 respectively) and add the two.This gives the answer as 5(5+9=14;1+4=5)

Or you could add the two numbers which will give you 68.The digital sum of 68 is also 5(6+8=14;1+4=5)

Additional note-For any number with any number of 9s and a few other digits,the digital sum can be calculated by ignoring the 9s.

That is to calculate the digital sum of 9698 I just need to add 6 and 8 to get the digital sum as 5.You can add all the 4 numbers and you will get the same answer.Try it..9+6+9+8=32 which also gives the digital sum as 5.

That is all for today.I will be coming out with more advanced concepts in further posts.Hope you all enjoyed reading the 2 fundae for today and that you come back for more.

A New Initiative!

I have had this idea in my head for a few days now to start a blog related to numbers.During my 1.5 years of teaching and getting taught for CAT,I think I have developed quite a few tips and tricks which should be very helpful to people as far as various MBA entrance exams are concerned.

Apart from my CAT prep I have always had a interest in numbers and have looked to develop my own ways of quickening calculations.

So here it goes.Watch out for some tricks that are sure to blow your mind..

Happy Reading..