Latest CAT Aptitude Question SOLUTION: What will be remainder when N=10^10+ 10^100 + 10^1000+ 10^10000..........+10^10000000000 is divided by 7

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(#M40037215) CAT Numerical Ability question Remainders Keep an EYE

What will be remainder when N=10^10+ 10^100 + 10^1000+ 10^10000..........+10^10000000000 is divided by 7

Asked In CAT Lovejeet Singh (10 years ago)

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Saraswathy  (10 years ago)

10^10+ 10^100 + 10^1000+ 10^10000..........+10^10000000000 % 7
Means 10^10%7 + 10^100%7 + 10^1000%7 + ...

10^1 % 7 = 3
10^2 % 7 = 2
10^3 % 7 = 6
10^4 % 7 = 4
10^5 % 7 = 5
10^6 % 7 = 1
10^7 % 7 = 3 [Again it goes on like 2,6,4,5,1,etc]

So cycle is upto the power of 6... So R[N % 7] is 6k+n...
10 => 6*1 + 4 ... R[10^10 % 7] = 4
100 => 6*16+4 ... R[10^100 % 7] = 4
1000 => 6*166+4 ... R[10^1000 % 7] = 4
So For all the powers of 10^(10^n), 4 is the remainder...
[4+4+4+4+4...+4]%7
10*4 % 7
40 % 7
5 % 7

Ans : 5

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Lovejeet Singh  (10 years ago)

Answer is damn right i.e. 5
I used the following method:
Acc to Fermat Little theorem Rem[M^(N-1)/N]=1 like Rem[10^6/7]=1
So here we have Rem[10^10/7]=>Rem[10^6*10^4/7]=>Rem[10^4/7]=>Rem[3^4/7]=4
Similarly other terms too give Remainder 4
Total Remainder=4*10 times=40
Rem[40/7]=5
Hence 5 Answer.
Bingo !!

Is this solution Helpfull? Yes (5) | No   

Aditi Bhite  (6 years ago)

if you use a calculator 10^10 , 10^100 , 10^1000 and so on all leave a remainder of 4 when divided by 7.
so the final remainder is the remainder of 4 * 10 by 7 = 5

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