What remainder does the expresion 10^10+100^100+1000^1000+ … [phone]^1000000000 leave when divided by 7?

Answer 1

Uhh... let's see...

Divisibility by 7 table, using powers:

If x is divisible by 7, then all subsequent powers will also be divisible by 7. If x is not divisible by 7, then here are the tables:

x^... 1 2 3 4 5 6
1.... 1 1 1 1 1 1
2.... 2 4 1 2 4 1
3.... 3 2 6 4 5 1
4.... 4 2 1 4 2 1
5.... 5 4 6 2 3 1
6.... 6 1 6 1 6 1

Since mod(10,7) equals 3, then we add 6 to "k", which is a variable we will use to track the divisibility.
Since mod(100,7) equals 2, then we add 4 to k, making k 10.
Since mod(1000,7) equals 6, then k becomes 11.
Since mod(10000,7) equals 4, then k becomes 15.
Since mod(100000,7) equals 5, then k becomes 18.
Since mod(1000000,7) equals 1, then k becomes 19.

Now, your 1000000000^1000000000 has nine zeros, so add 11 to 19 to make 30. Take the modulus of 30 and 7 to get your desired answer of 2.

Answer 2

Alexander was right, it is 0, although his explanation is lacking. Gimb's argument is missing one fundamental concept:

although 10 is 3 mod 7, if you want to calculate 10^10 mod 7, you must look at the exponent mod 6, not mod 7, since 10^6 is 1 mod 7 by Fermat's little theorem (or Legendre's Theorem from group theory). In fact, we find that 10 is 4 mod 6, and 10^2 is still 4 mod 6, so that any power of 10 is 4 mod 6.

If we examine any particular term (10^i)^(10^i) where 1 <= i <= 9, and use the fact that 10 is 3 mod 7 and any power of 10 is 4 mod 6, we find that (10^i)^(10^i) is congruent to (3^i)^(4) mod 7. But this is the same as 81^i mod 7, and that is the same as 4^i mod 7. Thus, your sum reduces to the sum for i going from 1 to 9 of 4^i mod 7, a geometric series, whose sum is (4^10-4)/3. But 4^3 is 1 mod 7, so 4^10 = 4*(4^3)^3 which is 4 mod 7, so 4^10-4 is 0 mod 7.

Alternatively, go to Mathematica and type:
For[tot=0;i=1, i<=9, i++, tot+=PowerMod[10^i,10^i,7]];Mod[tot,7] and you will get 0.

Answer 3

10 mod 7= 3

(1) 10^10 mod7 = 3^3 = -1
(2) = 2^2 = -3
(3) = -1^6 = 1
(4) = -3^4 = -3
(5) = -2^5 = 3
(6) = 1^1 = 1
no start over again with first : (7) = (1),(8)=(2) (9)=(3)
(1) + .. (9) = 2

Answer 4

I got 0. Basically I did the same as the last answer, but I had some of the answers different that were not recorded.

Answer 5

Just a bunch of stuff thrown together that has no answer.