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Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Tuesday, 5 February 2008, 06:13 PM
  Hi TG,
 I am hoping to start a a new quant thread for CAT2k8. I thought it would be better if we can proceed topic by topic. Right Now , I am working on Number Systems.So I thought of starting a new thread which will consist of all difficult number system problems. I hope we can make a difference in CAT by solving these difficult questions. So, Frm Today I am posting my doubts on number system. It's a request to those who will be posting answers , please do post the explanations also.


Regards,
Ansh
Re: Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Tuesday, 5 February 2008, 06:24 PM
  Hi, here comes my first sets of doubt.

1. A two-digit number is 18 less than the square of the sum of its digits. How many such numbers are there?

2. C is a Composite number with and even number of factors. Consider the follwong statement
a: C has a factor lying between 1 and sqroot(C)
b: C has a factor lying between sqroot(C) and 1

which of the following statements are true?

3.  If 25<=x<=49 and Y= [x^2 + 3sqroot(x)(x+9)+81]/[x+6*sqroot(x)+9], then what is the range of values of Y.

4.Let x , y and z be three natural numbers such that x+y+z=9m+10, where m is a natural number. For any m,which of the following holds true?
1. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34.
2.The maximum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34
3.The minimum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34
4. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34

5. N is a natural Number between 10 and 1000. P denotes the Product of the digits. S denotes the sum of the digits. If 6P+4S=4N, how amny values can N take?
Re: Quantz doubts for CAT2k8-Number System
by Little Star - Tuesday, 5 February 2008, 07:44 PM
  1. A two-digit number is 18 less than the square of the sum of its digits. How many such numbers are there?

if two digit no ab then 10a+b = a^2 + b^2-18 so 47 and 67 only possiblebig grin

5. N is a natural Number between 10 and 1000. P denotes the Product of the digits. S denotes the sum of the digits. If 6P+4S=4N, how amny values can N take?

9 value possible-16/26/36/46/56/66/76/86/96wink

Little Star
Re: Quantz doubts for CAT2k8-Number System
by TG Team - Tuesday, 5 February 2008, 07:35 PM
  5. N is a natural Number between 10 and 1000. P denotes the Product of the digits. S denotes the sum of the digits. If 6P+4S=4N, how amny values can N take?
Let N be a two digit number N = ab = 10a + b
P = ab
S = a + b
6ab + 4a + 4b = 40a + 4b
=> 6ab = 36a
=> b = 6 So total 9 two digit numbers i.e.  16,26,36,46,56,76,86,96.

Let N be a three digit number N = 100a + 10b + c
P = abc
S = a + b + c
6abc + 4a + 4b + 4c = 400a + 40b + 4c
=> 6abc = 396a + 36b
=> abc = 66a + 6b
=> b = 66a/(ac - 6) NO solutions
Only 9 values are there.smile

Re: Quantz doubts for CAT2k8-Number System
by Jim Braddock - Tuesday, 5 February 2008, 08:18 PM
  1. A two-digit number is 18 less than the square of the sum of its digits. How many such numbers are there?

let the no. is 10a+b.
so accord. to the prob. statement, 10a+b+18 = (a+b)^2
for a = 1 to 9, the L.H.S. will be, 28+b, 38+b, 48+b,.....,108+b.
out of above values, only 28+b, 48+b, 58+b, 78+b can be equal to perfect square 36, 49, 64, 81 respectively with the constraint that digit b can take integer value 1 to 9.
but only 78+b satisfies the given condition with digit b = 3.
so the no. is 63, as here a = 6.
which indicates that only one such 2 digit no. is possible.

Regards,
Jim
Re: Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Wednesday, 6 February 2008, 04:48 PM
  Buddy, Even I proceeded the same way but can u please explain how did u get there will be no solution for the  second case when there is a three digit number.

Rgds,
Ansh
Re: Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Wednesday, 6 February 2008, 04:49 PM
  Thanks for your answers but I wanted the solutions coz even I have the answerssmile
Re: Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Wednesday, 6 February 2008, 04:56 PM
  Thanks buddy for explaining me the trick
Re: Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Wednesday, 6 February 2008, 04:58 PM
  What about these questions? who can explain these questions?..I will shortly list my second set of doubt smile-

2. C is a Composite number with and even number of factors. Consider the follwong statement
a: C has a factor lying between 1 and sqroot(C)
b: C has a factor lying between sqroot(C) and 1

which of the following statements are true?

3.  If 25<=x<=49 and Y= [x^2 + 3sqroot(x)(x+9)+81]/[x+6*sqroot(x)+9], then what is the range of values of Y.

4.Let x , y and z be three natural numbers such that x+y+z=9m+10, where m is a natural number. For any m,which of the following holds true?
1. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34.
2.The maximum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34
3.The minimum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34
4. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34
Re: Quantz doubts for CAT2k8-Number System
by Little Star - Wednesday, 6 February 2008, 05:23 PM
  sorry dear in 5th ques u can find another 2 no.

Let N be a three digit number N = 100a + 10b + c
P = abc
S = a + b + c
6abc + 4a + 4b + 4c = 400a + 40b + 4c
=> 6abc = 396a + 36b
=> abc = 66a + 6b
=> b = 66a/(ac - 6)
now a from 1 to 9 , if ac=72 then ac-6=66 and b=a        so u can find another 2 no 998 & 889
No any other three digit can't satisfy because if 66 divided by 11 and 6, if ac=12 so then ac-6=6 so b=11a it is not possible and if ac=17 then ac-6=11 but 17 is prime so not possible
winkBingo!

Little Star

Re: Quantz doubts for CAT2k8-Number System
by TG Team - Thursday, 7 February 2008, 12:00 PM
 

4.Let x , y and z be three natural numbers such that x+y+z=9m+10, where m is a natural number. For any m,which of the following holds true?
1. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34.
2.The maximum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34
3.The minimum possible value of x^2 + y^2 + z^2 is 27m^2 + 60m + 34
4. The minimum possible value of x^2 + y^2 + z^2 is 27m^2 - 60m + 34

x + y + z = 9m + 10

and x2 + y2 + z2 = (x + y + z)2 - 2(xy + yz + zx) is minimumwhen (xy + yz + zx) is maximum.

And ( xy + yz + zx) is maximum when x = y = z = 3m + 10/3.

Substituting values for (x + y + z) as 9m + 10 and x, y and z

x2 + y2 + z2 minimum = 27m2 + 60m + 34. Option 3 is correct.smile

Re: Quantz doubts for CAT2k8-Number System
by Anshuman Bhar - Thursday, 7 February 2008, 02:37 PM
  Thanks buddy I got ur answer...but the thing is how to approach this kind of problem . I mean how to solve this equality b=66/(ac-6) so that we can find out the answer in a easier way?
Re: Quantz doubts for CAT2k8-Number System
by monica shah - Tuesday, 13 January 2015, 01:14 PM
  For 3 digit number
6abc=396a+36b
bc=66a+6(b/a)
So bc>=66
If(b,c)=(8,9) or (9,8) then b/a=1 so b=a
Therefore only two possibilities are 889 & 998
So in total there are 9+above 2= 11 possibilies.