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
Join The Discussion
Comments ( 4 )
Related Questions on Number System
Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The numbers are:
A. 12, 24, 36
B. 11, 22, 33
C. 12, 24, 32
D. 5, 10, 15
answer is 1
but remainder is 4
4*15/7=rem 1
Can anyone solve how 3^4/7 = 1 and 3^1000=2 in details??? otherwise the solution is really hard to understand.
How you are writing 3^4 from 3^25
10^1000 can be written as 3^250*4 right, then how could the remainder be 2?