A= 1¹⁰¹ +2¹⁰¹ +3¹⁰¹ + 4¹⁰¹ +......+34¹⁰¹. Find the remainder when A is divided by 70.

Hi,

This question can be easily solved by using binomial theorem.
Write 1 as 7-6 two as 7-5 and so on, remember that 7 multiple will be 7 and 8 will be written as 7+1 and so on.

Now when you will apply the BT for (7-1)^101 last term will be -1^101 which will be cancelled by (7+1)^101 s last term.

Similary these pairs will be formed for each of terms which will cancel eaqch other out except for multiples of 7.