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Number system :Remainder
by rakshit surana - Sunday, 8 July 2012, 08:37 AM
  What is the highest possible value of ‘n’ for which (3^1024) – 1 is divisible by 2^n?
(a) 13 (b) 10 (c) 11 (d) 12
Re: Number system :Remainder
by rakshit surana - Sunday, 8 July 2012, 12:30 PM
  kamal sir,
i followed 3^1024 = 3^(2^10)
now from option, taking 11..2^11, its euler function will be 2^11 * 0.5 = 2^10..
so 3^(2^10) = 1 mod 2^11
so, 3^(2^10) - 1 will be divisible by 2^11..
but it is incorrect..d is the ans???
PLZ HELP !!!!
Re: Number system :Remainder
by rakshit surana - Monday, 9 July 2012, 11:01 AM
  ques 2 .
How many 10-digit natural numbers in binary are there in which at least two 1’s come together?

kch hi samaj nhi aaya sad
Re: Number system :Remainder
by GadhoN Ka Baap - Monday, 9 July 2012, 12:15 PM
  Here, (3^1024)-1 = ((2+1)^1024)-1 = (1024C0 * 2^1024+....+1)-1.
i.e (1024C0 *2^1024+...+1024C1023 * 2)/2^n
and 1024C1023 * 2 =2^11. Hence highest value of n
will be 11.
Re: Number system :Remainder
by rakshit surana - Monday, 9 July 2012, 12:28 PM
  ans is 12 ?
Re: Number system :Remainder
by TG Team - Tuesday, 10 July 2012, 05:26 AM
 

Hi Rakshit smile

This question is exactly similar what I posted about three years back in an article on "Fibonacci Recursion". Slight difference is that now you are forming numbers by using the digits 0 and 1. So answer will nt be exactly same. Read the article and you'll be able to understand the funda. If you start making cases to solve this problem, then it's going to be a good lengthier question. I am pasting a picture of solved question from the article here.

Kamal Lohia 


Re: Number system :Remainder
by TG Team - Tuesday, 10 July 2012, 06:19 AM
 

Hi Rakshit smile

31024 - 1 = 9512 - 1 = (8 + 1)512 - 1 = 8512 + .... + C(512, 2)*82 + (512)8 + 1 - 1 = 21536 + ... + 511*214 + 212 = 212(4k + 1).

Hence 12 is the highest power of 2 which always divide the given number.

Kamal Lohia     

Re: Number system :Remainder
by rakshit surana - Tuesday, 10 July 2012, 08:18 AM
  thanx sir, forgot the basics,,urs is much simpler..
but sir i got one thing that if highest power was 12 then 2^11 will also leave remainder of 0 with the given number. so wats the problem with eULER HERE?? CAN U EXPLAIN
Re: Number system :Remainder
by TG Team - Tuesday, 10 July 2012, 09:12 AM
 

Hi Rakshit smile

There is no problem with Euler. phi(211) = 210.

So 32^10 = 1 mod 211

Or 32^10 - 1 is divisible by 211.

Kamal Lohia

Re: Number system :Remainder
by rakshit surana - Tuesday, 10 July 2012, 02:59 PM
  sir, one more
Find the remainder when (10^3+9^3)^752 divided by 12^12 ?
Re: Number system :Remainder
by TG Team - Tuesday, 10 July 2012, 04:36 PM
 

Hi Rakshit smile

That's a special case:

(103 + 93)752 = (123 + 1)752 = C(752, 3)*129 + C(752, 2)*126 + C(752, 1)*123 + 1} mod 1212.

I hope it is sufficient, you can very easily proceed from here. smile

Kamal Lohia 

Re: Number system :Remainder
by rakshit surana - Tuesday, 10 July 2012, 05:52 PM
  yes sir, got it ..thanx a lot
you reminded me a special property about 1729..12^3 + 1^3 = 10^3 + 9^3






















Re: Number system :Remainder
by rakshit surana - Wednesday, 11 July 2012, 06:29 PM
  sir , two more
1. A number N has three prime factors (2, 3, 5), it has 64 factors which are perfect cube and 27 factors which are perfect 4th power, then how may different values N can take ?

2. 9 points are marked on paper..wat is the maximum no of right triangle we can make from these points?
Re: Number system :Remainder
by sagar gupta - Wednesday, 11 July 2012, 08:20 PM
  For second question,i am getting 7 as the answer. is it correct?
Re: Number system :Remainder
by rakshit surana - Wednesday, 11 July 2012, 08:45 PM
  no..it is not ryt..i dont have ans..but it a two digit no.
Re: Number system :Remainder
by TG Team - Thursday, 12 July 2012, 09:46 AM
 

Hi Rakshit smile

2. It should be 16. Place the points symmetrically in L shape.

Re: Number system :Remainder
by rakshit surana - Thursday, 12 July 2012, 10:21 AM
  sir , i approach to make a2x2 grid..then in each 4 squares.. two diagonal bisect at right angle..then each square which give 4x4 = 16.. then in each of the 4 square only one diagonal can divide square in 2*2 = 4 ways..so total 4*4 = 16
then in big square of 2x2 ..again 2 diagonal make 4 rt triangle..and 1 diagonal make 2 rt traingle and other in d same way other diagonal in 2 ways..so total 8
overall 16+16+8 = 40
is it ryt sir?? more cases will be there?
and em unable to place 9 points in L shape..1 point remains mystery where to place it
Re: Number system :Remainder
by ajnabee sharma - Sunday, 28 July 2013, 05:06 PM
  12
Re: Number system :Remainder
by Navin Yadav - Sunday, 1 September 2013, 08:48 AM
  3^1024 -1=(3-1)(3+1)(3^2 +1)(3^4 +1)(3^8 +1)(3^16 +1)(3^32 +1)(3^64 +1)(3^128 +1)(3^256 +1)(3^512 +1)

All the terms are even. Second term (3+1)  is divisible by 4. Hence max. power of 2 contained in the above expression will be 12. smile cool
Re: Number system :Remainder
by MANISH Manish - Tuesday, 3 September 2013, 03:29 PM
  3^1024 - 1 = (2+1)^1024 - 1 = 2^1024 + .... + C(1024, 1022)*2^2 + (1024)2 + 1 - 1 = 2^11(k)
Approaching in dis way, why different answer is coming?
Re: Number system :Remainder
by Diwakar Tyagi - Friday, 11 October 2013, 01:03 AM
  I think it should be 36
First of all making all the possible rectangles and then 4 rt angles in each triangle
Re: Number system :Remainder
by sivasis dash - Wednesday, 15 July 2015, 11:21 AM
 

Hi Kamal,

But if I am doing by this way, say

31024-1=(2+1)1024-1

=1024C0*21024*10+........+1024C1023*21*11023+1024C1024*20*11024-1

=22048+.....+210*21+1-1

=22048+....+211

=211(some value)

both are done in the binomial way but y if I am taking this the highest power of 2 comes to be 11.

Re: Number system :Remainder
by sivasis dash - Wednesday, 15 July 2015, 11:26 AM
 

Hi navin,

how you concluded that the max power of 2 is 12 from the thing that 3+1 is divisible by 4.