Some Fine Problems  
I was fascinated when I saw a CAT preparation website named TotalGadha
for the first time. I downloaded almost every page of this MBA website on my pc and
made it a regular feature of my internet browsing. Slowly I started
participating in the forums, trying to answer the questions posed by
intellectual community of country and started getting appreciation in the form
of “Thank You” and smilies. I started wondering why I was addicted to this
site; why I was answering others’ questions on the forums. I got my answer
through TG himself in one of the threads where he said “I am compulsive problem
solver.” That sentence reflected me because that was true for me also. I took
the message of spreading the mathematical awareness to masses at almost no cost
from TG and started my journey towards CAT preparation community at my home
city, Hisar. And then came the real appreciation, may be which I
was looking for an invite from TG to join his team as he was to start his
classroom Avatar “Tathagat”. I wasn’t able to join him from inception because
of some commitments but now being a part of it is as challenging as rewarding. It is challenging because you need to tackle
challenges posed by largest pan India student community and rewarding because of those cute
gifts in the form of smilies given by the challenge posers. Idea is don’t run away from the challenges; because
if you aren’t able to solve them no one is going to blame you. But if you solve
it anyhow, fame and glory will follow. In any case you are at Profit. Coming to the point now, here I am explaining some
good questions which I have discussed in my classes at Tathagat. Some of these
questions we have used in our copyCATs and some may have been discussed in TG
forums also. I have catagorised the question in five categories and would like
to add to this repository from time to time for the well being of online
student community. It is highly recommended to try all of the listed problems by you yourself before peeping in the solution provided.
You might also like: CAT 2010 Quant Corner Some Mathematical Curios 
Re: Some Fine Problems  
Hi Rakesh, The honour is ours also. He is an amazing person to work with. Total Gadha 
Re: Some Fine Problems  
Hi Rakesh For me, there is no difference in the tag 'Bhai' or 'Sir'. As both show heart felt gratitudes of the other person only.

Re: Some Fine Problems  
Nice article 
Re: Some Fine Problems  
Kamal Sir? Totally Shocked........... 
Re: Some Fine Problems  
Hi SE If Gadha is TOTAL here then why shoudn't shock be TOTAL 
Re: Some Fine Problems  
Great job Kamal....thanx a lot... 
Re: Some Fine Problems  
Thanks Sir!!! In chocolates question, I too meant the same, forgot to explain it 
Re: Some Fine Problems  
SE? Where are you rey... Dagny and I were worried that you have quit preparation completely. No post, no mail, no msg... 
Re: Some Fine Problems  
Hi Boney 2^29 will give reminder 5 with 9. so sum of digits will be of the form 9K+5 = 9(4)+5 = 41 so 4 is missing. coz sum of 0+1+..9 = 45 If 9(3)+5 = 32 is not possible.

Re: Some Fine Problems  
thanks a lot!!!!!!!!!! really it makes gadha!!!!!!!!!!!!!! it's like morning exercise to brain!!!!!!!!! 
Re: Some Fine Problems  
Ans for Swimmer prob 20 times they will cross each other 
Re: Some Fine Problems  
in the problem n^2+4/n+5 , i am not able to understand the mesning of statement hcf not= to 1 means can u explain 
Re: Some Fine Problems  
y we divide by 8 only in quest x^22+ y^2+ z^2=1855 can u explain the complete method , please explain my querry soon because my paper is on 28 only 
Re: Some Fine Problems  
Thanks to read my mind Ankesh I feel really very happy to be of some help. Hope there is no grammatical error.

Re: Some Fine Problems  
Too good article sir! 
Re: Some Fine Problems  
Sir What do u mean by modulo 9?? 
Re: Some Fine Problems  
@TG sir/Kamal sir could you please check my answers for the questions which i have solved.. thankyou..

Re: Some Fine Problems  
KL plz expalin these too 
Re: Some Fine Problems  
Amit Just waiting for some more attempts

Re: Some Fine Problems  
Nidhi Where are these?? 
Re: Some Fine Problems  
@vivek You are rite.. i messed up finding the remainder.. Thanks mate 
Re: Some Fine Problems  
thank u Kamal Sir for solving the problem and making me understand that even 14*3=42 can become 52 in the back of the mind..here one for u too.... 
Re: Some Fine Problems  
Kamal sir, Thank you for such a wonderful article Given that 2^{2010} is a 606digit number whose first digit is 1, how many elements of the set S = {2^{0}, 2^{1}, 2^{2}, …, 2^{2009}} have a first digit of 4? The answer for this i think, should be 201. There is a pattern for first digits upto 2^10 and numbers repeat itself after every ^10 The first digits from 2^1 to 2^10 are as follows 2 4 8 1 3 6 1 2 5 1 and again from 2^11 to 2^20 the first digits are 2 4 8 1 3 6 1 2 5 1. So There should be (2010/10 *3 +1) digits ie 604 digits in 2^2010 and since from 2^1 to 2^10 there is only one 4 therefore there are 2010/10 = 201 fours in 2^2010 By solving with your method by taking 604 digits we get 201 also. 2010603*3 =201 Further similar patterns are seen in 3^1 to 3^21 which are again repeated from 3^22 as 3 9 2 8 2 7 2 6 1 5 1 5 1 4 1 4 1 3 1 3 1. in 4^1 to 4^5 as 4 1 6 2 1 and so on...... Sir, if i am wrong let me know where i went wrong. 
Re: Some Fine Problems  
thank a lot 
Re: Some Fine Problems  
Hey! Great problems.. But, few of them are of JEE standard not CAT!! 
Re: Some Fine Problems  
Leave those few Vijay 
Re: Some Fine Problems  
Nidhi Just returned after a few hectic days. Will check your responses soon and revert you back then. Till then there is no need to cry. 
Re: Some Fine Problems  
thnx kamal was waiting for reply

Re: Some Fine Problems  
did the same way : but my ans is worng so calculation prb : 
Re: Some Fine Problems  
Second now

Re: Some Fine Problems  
Third:

Re: Some Fine Problems  
HI KAMAL i could not make out your way of solution of last question.pls explain it a little. umesh 
Re: Some Fine Problems  
Thanks Umesh and JP for your kind words. Here is the fourth one:

Re: Some Fine Problems  
Taking yah example.. (3,4,5) 1) AB=3, BC=4 and CA=5 2) AB=4, BC=5 and CA=3 .. so on They all are congruent, but are they not different? This is what i wanted to ask. Sayonara 
Re: Some Fine Problems  
Q. a girl has to climb 10 steps. she takes either 2 steps or 1 step at a time. in how many ways can she climb these 10 steps? sorry i paste the problem here 
Re: Some Fine Problems  
one more bro....
Find the positive integral solutions of equation
x1*x2*x3*x4=120 umesh 
Re: Some Fine Problems  
S = (3 + 3^2 + 3^3 + … + 3^400) – (7 + 7^2 + 7^3 + … + 7^201). last two digits of this sum? 
Re: Some Fine Problems  
you did sy any thing about my solution of unit digit question? BTW, extremely thanks for these wonderful solutions um 
Re: Some Fine Problems  
ONe more........... In how many ways can 8 persons sit at a round dining table in such a manner that all will not have the same neighbours in any 2 arrangements? 
Re: Some Fine Problems  
pls kamal look into my last question solution...... give your opinion? since i am a bit sceptical about that?

Re: Some Fine Problems  
thanks brother!!!!!! i knew ....you would .... umesh 
Re: Some Fine Problems  
hi problem solver one more If a,b,c are natural numbers , then how many number of unordered (a,b,c) satisfy 
Re: Some Fine Problems  
ek aur sar dard(headache) If r, s, t are prime numbers and p, q are positive integers such that LCM of p, q is r^2*t^4*s^2, then the number of ordered pair (p, q) is 
Re: Some Fine Problems  
Hats off.....brother right brother...plsssssssssssssssssssssssss explain. hey .... you are 'PANACEA' for me. umesh

Re: Some Fine Problems  
hi kamal, it is me brother.how come you know the person.BTW , i am overwhelmed with your logical skill.thanks again umesh 
Re: Some Fine Problems  
hi Kamal i did not have the idea brother. i will never do it again. umesh 
Re: Some Fine Problems  
Hi Umesh It's good to see you promoting our (I am including you too) efforts to the mass for their own benefit. That's really a act of benevolence.

Re: Some Fine Problems  
hi bro... kaise karenge? Finding last nonzero digit of 8293! 
Re: Some Fine Problems  
i have not heard you since today morning .......probably enjoying ....'3 idiot,

Re: Some Fine Problems  
thanks kamal .... i had problem in verfying for 11. when you have time ,just see two other problem as well. regards umesh

Re: Some Fine Problems  
Hi Umesh I used to do this type of question by a bit lengthy way. But your large number (8293!) has made me think alternatively.
From a block of every 10 consecutive numbers starting from 1 (as above), we have two numbers which are multiple of 5 and when multiplied by two 2’s they will generate two zeroes. Now leaving the multiples of 5 intact, I remove the two 2’s from the remaining eight numbers from same block of 10 numbers. Taking a general block, I have the product as (10x + 1)(10x + 2)(10x + 3)(10x + 4)(10x + 5)(10x + 6)(10x + 7)(10x + 8)(10x + 9)(10x + 10) Or (10x + 1)(10x + 2)(10x + 3)(10x + 4)(10x + 6)(10x + 7)(10x + 8)(10x + 9) leaving two multiples of 5 aside. Or 100x^{2}(4a) + 10x(4b) + 1×2×3×4×6×7×8×9 Or 100ax^{2} + 10bx + 18144 taking two 2’s aside. So finally I am having unit digit of a block of 10 consecutive numbers as 4 (leaving multiples of 5 and taking out two 2's) Now coming to the original question: Last nonzero digit of 8293! = Last nonzero digit of 8290!×1×2×3 = 4^{829}×6×Last nonzero digit of 1658! = 4×2×4^{165}× Last nonzero digit of 330! = 2×4^{33}×Last nonzero digit of 66! = 8×6×4^{6}×last nonzero digit of 12! = 8×2×4×2 = 8. So the last nonzero digit of 8293! is 8. You can verify it here also. http://www.wolframalpha.com/input/?i=8293%21+mod10^2071&t=ietb01 
Re: Some Fine Problems  
@ Kamal. for TGites, u r Rancho of 3 Idiots. 
Re: Some Fine Problems  
Umesh Can you elaborate that ∑6 part?

Re: Some Fine Problems  
hi kamal wishing you very^(100000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000) happy new year umesh

Re: Some Fine Problems  
Answer to Fifth question now: Though it is quite late still quest for knowledge never ends.

Re: Some Fine Problems  
yaar ... see the question and help Let m and n be positive integers such that1=<m<n. In their decimal representations, the last three digits of1978^m are equal, respectively, to the last three digits of1978^n. Find m and n such that n+m has its least value. 
Re: Some Fine Problems  
Amazing article.... ur too good.. Btw I did my part in clicking on that share link Cheers 
Re: Some Fine Problems  
Hi Siddharth You have missed the digit 4. 
summation of reciprocal  
hi please suggest and explain the logic of the following series 1+1/2+ 1/3 +1/4 + ........+ 1/50 
Re: Some Fine Problems  
sir we cannot get the 2009 as sum when 334 dice are rolled, the maximum value is 2004 then probability is zero i didnt get the 7a1........... moreover it cant be less than 335 
Re: Some Fine Problems  
great site...great questions..thankss they are really helpful 
Re: Some Fine Problems  
Can anyone pls explain the solution : 13a + 17b <221 How 13(a'+1) + 17(b'+1) =221 and (171)(131)1 came into picture? 
Re: Some Fine Problems  
hi Kamal Can u pls elaborate on {16^{2}/2(1)(2)(3)} = {256/12} = 21. How to solve these type of problems? x + 2y + 3z = 16 or x + 2y + 3z <= 16 Thanks 
Re: Some Fine Problems  
Hii Kamal Sir, Can U please explain what does m=n (mod x) means??? How do we use it in number system problems??? 
Re: Some Fine Problems  
kindly solve dis question plzzzzzzz a 12 digit number is divisible by 72 consist of 4 and 6 only. find the no. of 6(largest possible number) 
Re: Some Fine Problems  
666 444 444 444 
Re: Some Fine Problems  
Sir, 666 444 444 444 is not divisible by 8, it s divisible by 9, though. Don't you think the answer should be 664 444 444 464? because we are looking for a no. divisible by 72? 
Re: Some Fine Problems  
Hi Kamal You are right. Number should be 664 444 444 464. But question was asking about the number of 6's in the number that should be 3 only. Kamal Lohia 
Re: Some Fine Problems  
i get addicted to totalgadha ....ur post just mesmerised me....last three digits of 2988^678 ..what will be its easy technique? 