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The remainder when 1010 + 10100 + 101000 + . . . . . . + 101000000000 is divided by 7 is

A. 0

B. 1

C. 2

D. 5

E. 4

Answer: Option B

Solution(By Examveda Team)

Number of terms in the series = 10.
(We can get it easily by pointing the number of zeros in power of terms.
In 1st term number of zero is 1, 2nd term 2, and 3rd term 3 and so on.)
$$\frac{{{{10}^{10}}}}{7},$$   Written as, $$\frac{{{{\left( {7 + 3} \right)}^{\left( {4 \times 2 + 2} \right)}}}}{7}$$
The remainder will depend on $$\frac{{{3^2}}}{7}$$
So, remainder will be 2
$$\eqalign{ & \frac{{{{10}^{1000}}}}{7},\,{\text{remainder}} = 2 \cr & \frac{{{{10}^{10000}}}}{7},\,{\text{remainder}} = 1 \cr} $$
So, we get alternate 2 and 1 as remainder, five times each.
So, required remainder is given by
$$\eqalign{ & \frac{{\left( {2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1} \right)}}{7} \cr & = \frac{{15}}{7} \cr} $$
Remainder when 15 is divided by 7 = 1

This Question Belongs to Arithmetic Ability >> Number System

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Comments ( 4 )

  1. Sahid Alom
    Sahid Alom :
    4 years ago

    answer is 1
    but remainder is 4
    4*15/7=rem 1

  2. Nazia Ashraf
    Nazia Ashraf :
    5 years ago

    Can anyone solve how 3^4/7 = 1 and 3^1000=2 in details??? otherwise the solution is really hard to understand.

  3. Anusree Nandy
    Anusree Nandy :
    7 years ago

    How you are writing 3^4 from 3^25

  4. Sourajya Kumar
    Sourajya Kumar :
    8 years ago

    10^1000 can be written as 3^250*4 right, then how could the remainder be 2?

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