Mind Blowing Questions: Season 2

[amar_kashyap]

Hi Folks
I propose to prepare a list of problems from various quant threads which are of CAT or higher standard. I will keep on editing the list so that we have a good collection of problems in one place rather than scattered here and there. I think tht's okay with you all. So here we go.

The basic intention is to collect all good problems from various quant threads and try to solve them one by one. We begin with the "mind blowing questions thread" of the last year. Lot of questions in that thread remained unsolved .We will take up the unsolved as well as the solved ones and come up with the solutions. The best and shortest solution will be compiled together for future reference. Here also we will be posting questions with no for easy posting and record keeping..

We will take up a next question only when the previous problem has been solved as far as possible.Only when we are really tsuck up we will move ahead with the next question..

[Govi]

@amar_kashyap you have already posted that compilation on the MB Thread and then another thread for future compilations . Most of the problems you have posted from MB have been solved on that thread. Infact some of them have been solved in the most beautiful way possible.

If you intend to discuss these problems Continue in the MB thread itself. There are many unsolved problems on the previous pages, even these could be taken up there. The Quant Marathon thread could also be utilized for the same.

PS: If you intend to start MB Season 2 then the thread name can be edited accordingly.

[amar_kashyap]

So here goes the first question: I am planning to move backward on the MB thread of previous year from 50th page to 1st page.

problem #1
What is tens digit in 32^32?

[the_CONSTANTINE]

hmm..

Is the Answer 7.
Seems to be a pretty decent question..Lets see...The question is not the last digit(which can be effortlessly found out to be 6)....But the ten's digit!

32^32, can be written as (2^5)^32..which is 2^160=

2^12=4096=-4mod 100
2^12^6=2^72=-4^6=4096=-4mod100
2^72^2=16mod100=2^144
2^144*2^12=2^156=16mod100*(-4)mod100
2^156= -64mod100
2^156*2^4=2^160= -64*16mod 100=-1024mod100
Which is nothing but -24mod100
which is 76...
Hence Ten's Digit is 7...

Method is long....Hoping that some1 gives a better n fast solution!!

Cheers

[amar_kashyap]

Originally Posted by the_CONSTANTINE

hmm..
Seems to be a pretty decent question..Lets see...
32^32, can be written as (2^5)^32..which is 2^160=
....Ten's digit is nothing but remainder after division by 100...!!
One can check that approach as well...Hope I am correct..!!

Cheers

The approach is fine. What is the answer... Plz post the full solution..

[reachmonil]

Thread renamed.
Let the proceedings begin!

PS: Juz take care that the level of Qs in the 'Mind Blowing Qs 1 (mind blowing questions....@quant)' thread is maintained, which I beleive is a tough ask. And, kudos to the regulars outta there, Aarav, anupam, khanna_summit, Apple, etc. for their contribution!

[amar_kashyap]

Originally Posted by the_CONSTANTINE

hmm..

Is the Answer 7.
Seems to be a pretty decent question..Lets see...The question is not the last digit(which can be effortlessly found out to be 6)....But the ten's digit!

32^32, can be written as (2^5)^32..which is 2^160=

2^12=4096=-4mod 100
2^12^6=2^72=-4^6=4096=-4mod100
2^72^2=16mod100=2^144
2^144*2^12=2^156=16mod100*(-4)mod100
2^156= -64mod100
2^156*2^4=2^160= -64*16mod 100=-1024mod100
Which is nothing but -24mod100
which is 76...
Hence Ten's Digit is 7...

Method is long....Hoping that some1 gives a better n fast solution!!

Cheers

Every Eqality sign prcedes division with 100 except the first one.

32^32=(1024)^16=(24)^16= (576)^8=(24)^8=(24)^4=24^2=576 .Hence second last digit=7.

[amar_kashyap]

problem#2
Prove that the number which when in decimal system is expressed by means of 91 unities is a composite number.

In between if there are other methods of doing problem#1...Please post

[amar_kashyap]

Where is quant Janta.. Everbody sleeping or what?
C,mmon guys.. Start blowing your minds .....

[khanna_sumit]

Originally Posted by amar_kashyap

problem#2
Prove that the number which when in decimal system is expressed by means of 91 unities is a composite number.

In between if there are other methods of doing problem#1...Please post

1111111 is one of the factors, so is 1111111111111