Wats the remainder when 10^10 + 10^100 + 10^1000......+10^10000000000 is divided by 7??(do without calculator)

Question

pls explain the process...dont take help of calculator and do in quick time

Answer 1

5.

Note that 10,000,000,000 ≡ 4 mod 7 (you can work this out by simple long division). Note also that 4^10 ≡ 4 mod 7 (this can be found quickly using exponentiation by squaring). Thus, if x≡4 mod 7, so is x^10, x^100, x^1000... and so on. But since all of the above numbers are (10^10)^(10^k), where k is some integer, all of the above numbers are congruent to 4 mod 7. There are 10 such numbers, so their sum is congruent to 10*4 mod 7, which is 5.

Edit: sorry, I messed up the properties of exponents on my first posting. Proof is valid now.

Answer 2

100 Divided By 10

Answer 3

10 divided by 7 remainder 3
10^3 divided by 7 remainder 3^3 or 27 or 6 or -1
Now:
10^10 divided by 7 remainder (-1)^3*10 or -10 or -3 or 4
10^100 divided by 7 remainder (10^3)^33*10 or -10 or 4
10^1000 divided by 7 remainder (10^3)^333*10 or -10 or 4
...
10^10000000000 divided by 7 remainder (10^3)^3333333333*10 or -10 or 4
So total we have
10^10 + ... + 10^10000000000 divided by 7 remainder 4*10 or 40 or 5
My answer: 5

Answer 4

10=3mod7
further
10^10=3^10mod7
3^10=-3mod7
3^100=-3^10=-3mod7
and so on

hence required sum S is
(-3+-3+-3..10 times)mod 7
or -30 mod 7
or 5mod7

Hence remainder 5.