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The remainder when 1010 + 10100 + 101000 + . . . . . . + 101000000000 is divided by 7 is
Answer & Solution
Correct Answer:
Option
B
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
(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
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LoginBECAUSE 3^6 divide by 7 remainder 1.
So, remaining 3^4 divide by 7 remainder 4.
Remainder 4 comes 10 times ( 10^10 to 10^1000000000)
Then, 40/7 rem is 5
but remainder is 4
4*15/7=rem 1