The remainder when 10¹⁰ + 10¹⁰⁰ + 10¹⁰⁰⁰ + . . . . . . + 10^1000000000 is divided by 7 is

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 1^st term number of zero is 1, 2^nd term 2, and 3^rd 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