CAT 2008: Quantitative Questions a Day 1 to 50 - The discussions

[srikar2097]

Originally Posted by sabsebadapaagal

didnt get your approach .... lets consider the case when N= 33 and N=22 the both above no have factors which are prime nos.... more over when srikar knows that last digit is 3 the only possible soln it 33... similarly in case of 22..

pls let me know if I have missed something or made some mistake ....

Boss! If srikar knows only the units digit, say 3 (which is prime). then how can he say for sure that the number N will be divisible by atleast 2 primes? It's not definite right? For all you know the number could be 13. Did you get it now?

[rajeev_hts]

We know that Srikar knows only unit digit and by this knowledge he is able to find that number is divisible by atleast 2 primes. this is possible with unit digit 8 an 0.

Then arbit_rageur knows the number by only having 2 pieces of knowledge:
1. Number of divisors
2. At least 2 divisors are prime.

Now we know following things:

1. Number of divisors can not be prime.
2. Number of divisors should be at least 4 because for unit digit 8 possible numbers having 2 prime factors are 18, 28, 38......98, 10, 20, 30..50

if there are 4 divisors then arbit can infer following numbers :- 6, 10, 14,15,21, 22, 26,38 .....
if there are 6 divisors then possible values are 18, 20, 28, 68...
if there are 8 divisors then possible values are, 30, 42, 24, 40, 78 ..
if there are 9 divisors then possible value is 36 (Unique)
if there are 10 divisors then possible values of N is 48 (unique)
If the number of divisors are more than 10 no possible value of N (<60).

so we arbit left is with 2 unique values 36, 48.

but arbit knows that last digit shoul be 8 as srikar told befor him that possible number has at keast 2 factor.

then arbit will know it is 48. and he says he knows the value.

Srikar knows that if arbit can come to some unique number then it should be only 48 because as per Srikar unique number of divisors comes from these numbers only.

so answer is (b) 1 and N is 48.

[selebratinglife]

Originally Posted by Aarav

This is an easy question but interesting.

Two budding MBAs who are also mathematicians, Srikar and arbit_rageur, play a game. The computer selects some secret positive integer N < 60 (both Srikar and arbit_rageur know that , but that they don't know what the value of N is). The computer tells Srikar the unit digit of N, and it tells arbit_rageur the number of divisors of N. Then, Srikar and arbit_rageur have the following dialogue:

Srikar: I don't know what N is, and I'm sure that you don't know either. However, I know that N is divisible by at least two different primes.

arbit_rageur: Oh, then I know what the value of N is.

Srikar: Now I also know what N is.

Assuming that both Srikar and arbit_rageur speak truthfully and to the best of their knowledge, how many possible values of N are there?

(a)0 (b) 1 (c) 2 (d) 3 (e) none of these

I have a fundamental doubt here aarav,
If arbit and srikar claims they know N, does it mean that they deciphered uniquely what 'N' is? Then the question of how many possible values of 'N' are there doesn't hold good..And if what they say is correct(Also assuming they haven't done any mistake in calculating) I can't find any number which can be deciphered just knowing the factors and the fact that it is divisible by atleast two primes..
Say 2^5 is the max power below 60 which has factors 6..It can be achieved by 4*3,4*5,4*7,4*11..

Pardon my ignorance and please clarify..

[nbangalorekar]

Originally Posted by Aarav

This is an easy question but interesting.

Two budding MBAs who are also mathematicians, Srikar and arbit_rageur, play a game. The computer selects some secret positive integer N < 60 (both Srikar and arbit_rageur know that , but that they don't know what the value of N is). The computer tells Srikar the unit digit of N, and it tells arbit_rageur the number of divisors of N. Then, Srikar and arbit_rageur have the following dialogue:

Srikar: I don't know what N is, and I'm sure that you don't know either. However, I know that N is divisible by at least two different primes.

arbit_rageur: Oh, then I know what the value of N is.

Srikar: Now I also know what N is.

Assuming that both Srikar and arbit_rageur speak truthfully and to the best of their knowledge, how many possible values of N are there?

(a)0 (b) 1 (c) 2 (d) 3 (e) none of these

since Srikar can tel that the no. is divisible by atleast 2 primes, the last digit shud be 0.
now the different cases arise:
50-6 divisors
40-8 divisors
30-8 div.
20-6 div.
10-4 div.
since arbit rageur can definitely determine the no. it must be 10.
so ans is 'b' only 1 possible value

[srikar2097]

Originally Posted by selebratinglife

I have a fundamental doubt here aarav,
If arbit and srikar claims they know N, does it mean that they deciphered uniquely what 'N' is? Then the question of how many possible values of 'N' are there doesn't hold good..And if what they say is correct(Also assuming they haven't done any mistake in calculating) I can't find any number which can be deciphered just knowing the factors and the fact that it is divisible by atleast two primes..
Say 2^5 is the max power below 60 which has factors 6..It can be achieved by 4*3,4*5,4*7,4*11..

Pardon my ignorance and please clarify..

By knowing just the units digit, srikar is able to tell that N is divisible by atleast 2 primes. This is a big clue...

[Aarav]

Originally Posted by selebratinglife

I have a fundamental doubt here aarav,
If arbit and srikar claims they know N, does it mean that they deciphered uniquely what 'N' is? Then the question of how many possible values of 'N' are there doesn't hold good..

Pardon my ignorance and please clarify..

You have a valid point -> the answer will be choice (b) only but we have to find that value of N here as N is unique.

[nbangalorekar]

Originally Posted by neynetr

As 16 is not possible so i guess we can rule out 6 as units digit because just by looking at the last digit srikar is able to say that it has at least two prime factors...

so we'r left only with 10,20,30,40 and 50... out if which only 10 has uniqe no of factors.... so answer should be only one value of N... (B).... ( still the same)

However i think therz a catch in the question...

Thru srikar, arbit comes to know that the number has two or more prime factors.. He still doesnt know of the 10 to 50 thing as srikar has not told him that he has come to know of two prime factors by looking at the last digit of the number (this information is limited to srikar only..... right..???)
so now arbit has a list of all the numbers <60 to choose from for which the no of prime factors is two or more than two....

Is my observation correct....???

dude only for unit's digit '0' can you say that the no.has atleast 2 prime factors. for other digits you can have exceptions. if nothing else since the no. is positive it has to be >0. so it can also be a single digit no. in which case there is no chance of it having >2 prime factors. (even for 6, the no. can be 16 where it will have just one prime factor)
but since the no. is >0, the only possible case is the unit's digit is 0.

[rajeev_hts]

Originally Posted by rajeev_hts

We know that Srikar knows only unit digit and by this knowledge he is able to find that number is divisible by atleast 2 primes. this is possible with unit digit 8 an 0.

Then arbit_rageur knows the number by only having 2 pieces of knowledge:
1. Number of divisors
2. At least 2 divisors are prime.

Now we know following things:

1. Number of divisors can not be prime.
2. Number of divisors should be at least 4 because for unit digit 8 possible numbers having 2 prime factors are 18, 28, 38......98, 10, 20, 30..50

if there are 4 divisors then arbit can infer following numbers :- 6, 10, 14,15,21, 22, 26,38 .....
if there are 6 divisors then possible values are 18, 20, 28, 68...
if there are 8 divisors then possible values are, 30, 42, 24, 40, 78 ..
if there are 9 divisors then possible value is 36 (Unique)
if there are 10 divisors then possible values of N is 48 (unique)
If the number of divisors are more than 10 no possible value of N (<60).

so we arbit left is with 2 unique values 36, 48.

but arbit knows that last digit shoul be 8 as srikar told befor him that possible number has at keast 2 factor.

then arbit will know it is 48. and he says he knows the value.

Srikar knows that if arbit can come to some unique number then it should be only 48 because as per Srikar unique number of divisors comes from these numbers only.

so answer is (b) 1 and N is 48.

Answer is (b) But the value should be 10. I missed the point if unit digit is 8 then 8 itself can't have 2 prime divisors.

10 has 4 divisors that is unique for a number with unit digit 0.

[Aarav]

Originally Posted by nbangalorekar

since Srikar can tel that the no. is divisible by atleast 2 primes, the last digit shud be 0.

Let me try to test more on this.

OK -> Assume, the number is a 2 digit and N < 100. Can we now have a unique N?

I will go a step further and talk about a famous problem.

Two budding MBAs who are also mathematicians, Srikar and arbit_rageur, play a game. The computer selects two secret positive integer x, y such that x, y < 20 (both Srikar and arbit_rageur know that , but that they don't know what each number is). The computer tells Srikar the sum x+y, and it tells arbit_rageur the number xy . Then, Srikar and arbit_rageur have the following dialogue:

Srikar: I don't know what product xy you have, and I'm sure that you don't know either what sum x+y I have.

arbit_rageur: Oh, then I know what the value of x+y is.

Srikar: Now I also know what xy is.

Assuming that both Srikar and arbit_rageur speak truthfully and to the best of their knowledge, what is x and y here?

[pallaviagrawal]

since srikar knows the units digit,and concludes that the no has atleast 2 prime divisors ,we know that the units digit can be 0,4 or 8(not 1,2,3,4,5,6,7,8,9 because they need not have 2 prime divisors..nos ending with an odd digit might be prime ;2,4,6,8 have possible values of 32 and 4,16,8 which dont have more than 1 prime factor).

0-10(4),20(6),30(,40(8 ),50(6)

(the no inside the bracket indicates the no of divisors)

from the conversation betweent the two we conclude that the no has distinct no of divisors.
so the no is 10.
n has only one possible value