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regarding number system problems
by tribhuvan kumar sangerpal - Monday, 7 April 2008, 01:38 AM
 

Dear TG PLEASE HELP ME IN SOLVING THESE PROBLEMS

1.what is the remainder when 128 ^ 1000 is divided by 153.

2.what is the remainder when 50 ^ 51^52 is divided by 11

3.what is the remainder when 32 ^ 33^ 34 is divided by 11

4.what is the remainder when 30^72^87 is divided by 11

5. what is the remainder when 50^56^52 is divided by 11

6.what is the remainder when 33^34^35 is divided by 7

7.what is the remainder of 2(8!)-21(6!) +14 (13!)

8. find 28383 term of series 123456789101112..............

 

 

Regards

Tribhuvan

 

Re: regarding number system problems
by TG Team - Monday, 7 April 2008, 05:51 AM
  2.what is the remainder when 50 ^ 51^52 is divided by 11
As φ(11) = 10
=> 5010k = 1mod11
So 5152 is to be written in the form of 10k + a.
Now unit digit of 5152 = 1 => 5152 = 10k + 1.
=> 5051^52 = 50(10k + 1) = 50mod11 = 6mod11smile

3.what is the remainder when 32 ^ 33^ 34 is divided by 11
As HCF(32,11) = 1 we can use same method as above.
That is 3334 is to be written in the form of 10k + a.
But unit digit of 3334 = 9 => 3334 = 10k + 9.
=> 3233^34 = 32(10k + 9) = 329mod11 = (-1)9mod11 = (-1)mod11 = 10mod11smile

4.what is the remainder when 30^72^87 is divided by 11
Going by same method, 7287 = 10k + 8
=> 3072^87 = 30(10k + 8) = 308mod11 = (-3)8mod11 = 33mod11 [ 35 = 1mod11]
= 27mod11 = 5mod11.smile

5. what is the remainder when 50^56^52 is divided by 11
5652 = 10k + 6
=> 5010k + 6 = 506mod11 = 56mod11 = 5mod11smile
Re: regarding number system problems
by ambar patil - Monday, 7 April 2008, 01:18 PM
 

8 . It is an AP with a=1 , d=1  . So T28383 = 1 + ( 28383 - 1 ) 1 = 1 + 28382 = 28383 .

6.  Since HCF ( 33 , 7 ) is 1 , we can use Euler’s method to solve this .Now φ(7) = 6 . So Remainder [ 33mod 7  ] = 1 .

So we try to express 3435 in the form 6k+a . Now we find what is the remainder when 3435 is divided by 6 . 

34 and 6 have a factor in common i.e 2 . so we take out this common 2 and find  remainder when 1735 is divided by 3 . φ(3) = 2 .

Therefore 1735  mod 3 = 17 34 * 17 mod 3 = 1 * 2 = 2 . Since we had taken out the common factor 2 , we now multiply the remainder by 2 which gives us remainder of 4 .

so i can write  3435 is mod 6 = 4 or 3435 = 6K+4  .

Now 33^3435 mod 7 = 33 6k+4 mod 7 = 33 4 mod 7 =  2 .

Please let me know if 2 is the correct answer .

Re: regarding number system problems
by TG Team - Monday, 7 April 2008, 06:36 PM
 

 1.what is the remainder when 128 ^ 1000 is divided by 153.

Ist method:

1281000 = 27000 = 2(96*72 + 88) = 288mod153.

[As φ(153) = 96 and 296 = 1mod153]

As also 27 = -25mod153

=> 288 = 2(7*12 + 4) = 24*2512mod153 = 24*136 = 24*163 = 216 = 2(7*2 + 2) = 4*252 = 4*13 = 52mod153.smile

[As 252 = 13mod153 and 132 = 16mod153]

 

IInd method:

153 = 32*17 = 9*17

and 1281000 = 21000mod9 = 2(3*333 + 1)mod9 = -1*2mod9 = 7mod9.

also 1281000 = (-8)1000mod17 = 23000mod17 = 24*750mod17 = 1mod17.

Also 9*2 - 17*1 = 1

Using Chinese Remainder Theorem

1281000 = (9*2*1 - 17*1*7)mod153 = -101mod153 = 52mod153.smile

Re: regarding number system problems
by Small Wonder - Monday, 7 April 2008, 08:40 PM
 

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Small

Re: regarding number system problems
by bimal mohan - Monday, 7 April 2008, 11:36 PM
 

hi  ambar,

    2 is the correct  ans.

Rem   [33^34^35 / 7]   =  Rem   [33^4 / 7]    as   Using FLT,   Rem   [34^35/ 6] 

                                                                                            =  2 Rem[2^34 * 17^35/ 3] 

                                                                                            =   2 * 2  = 4

now, Rem   [33^34^35 / 7]   =  Rem   [33^4 / 7]  = Rem   [( -2)^4 / 7]   =2

Re: regarding number system problems
by shweta sahu - Thursday, 8 May 2008, 04:59 PM
  where can we get the solutions for the no system problems  quiz in TG
Re: regarding number system problems
by sunny R - Saturday, 14 April 2012, 06:54 PM
  hey!
can u plz explein this last step?i m unable to understand.
1281000 = (9*2*1 - 17*1*7)mod153 = -101mod153 = 52mod153
thnx in advance
Re: regarding number system problems
by TG Team - Sunday, 15 April 2012, 12:42 PM
 

Hi Sunny smile

This is simply application of Chinese Remainder Theorem.

For two divisors D1 and D2 such that HCF(D1 , D2) = 1, first find integers x, y so that xD1 + yD2 = 1

Next let N r1 mod D1

and also N r2 mod D2

Then according to Chinese Remainder Theorem (CRT):

N ≡ (xD1r2 + yD2r1) mod D1D2.

Replece the variables with numbers and try to use this in two three examples. I hope you can get it easily. smile

Kamal Lohia

Re: regarding number system problems
by Rakesh Purohit - Monday, 18 June 2012, 05:13 PM
 

8..Hi ambar.This is asking about the 28383 term and not the number i guess.

see from [1-9]-it is 9 digits

              [10-99]it is 180 digits.

              [100-999] it is 2700 digits.

So if we add all we will get-2889 digits.

Now substracting from 28383 we will get 25494 deigits.

Means we need 25494 more digits to reach at 28383 term.

All of the 25494 digit will belong to [1000-9999].

so dividing 25494/4 we get 6373 as the output and 2 as the reminder.

so frm 1000 if we start we will reach at 7372 as the 3873th number.and the next number will be 7373 in the series.

so for reminer if we take the first 2 digits of 7373 we can find the 28373th digit.

so my answer is-3

 

 

Re: regarding number system problems
by gadha random - Saturday, 1 June 2013, 01:19 AM
  2.what is the remainder when 50 ^ 51^52 is divided by 11
As φ(11) = 10
=> 5010k = 1mod11
So 5152 is to be written in the form of 10k + a.
Now unit digit of 5152 = 1 => 5152 = 10k + 1.
=> 5051^52 = 50(10k + 1) = 50mod11 = 6mod11



As φ(11) = 10
I could not understand what this means. please explain
Re: regarding number system problems
by TG Team - Sunday, 2 March 2014, 05:17 PM
  Hi

I know I am late to respond but I hope it'll be beneficial for newcomers.
Here
φ(11) = phi(11).
It is basically because of symbol recognition flaw by the website.

I hope you know that phi(n) is Euler's Totient Function which is used to find remainders of the large numbers raised to some exponents.

Kamal Lohia
Re: regarding number system problems
by Aniruddha Ghosh - Monday, 28 July 2014, 02:28 AM
  As φ(11) = 10.
how to calculate this value???please reply
Re: regarding number system problems
by Aniruddha Ghosh - Monday, 28 July 2014, 02:30 AM
  no sir i really dont know how to calculate phi(n) because i'm new to this field..can u tell me once again with an example say phi(7) please
Re: regarding number system problems
by Aniruddha Ghosh - Monday, 28 July 2014, 02:35 AM
  sir i really dont know how to calculate phi(n) because i'm new to this field..can u tell me once again with an example say phi(7) please
Re: regarding number system problems
by TG Team - Monday, 28 July 2014, 09:53 AM
  Hi Aniruddha smile

Please go through our lesson on Remainders.

I hope this will help you. smile

Kamal Lohia
Re: regarding number system problems
by Sayan Das - Tuesday, 16 September 2014, 03:40 PM
  Why cant we apply Eulers theorem in the previous sum?? Even there the hcf of two nos is 1
Re: regarding number system problems
by Anand Singh - Wednesday, 15 July 2015, 01:09 AM
  Why are we dividing by 10 only ?