Compute 10^(10^(100)) Mod 7.?

Question

please list the steps on how to do this problem. Thnx

Answer 1

One of the rules of modular arithmetic is that for any prime p and any integer a, a^(p-1) ≡ 1 (mod p). (This is a consequence of Fermat's Little Theorem, if you want to look into it a bit more.)

So 10^6 ≡ 1 (mod 7), and therefore 10^(6k) ≡ 1 (mod 7) for any integer k. So if we can write 10^100 = 6k + r for some integers k and r, we know 10^(10^100) = 10^(6k).10^r ≡ 10^r (mod 7). So we need to find an r in {0, 1, ..., 5} such that 10^100 ≡ r (mod 6).

Now 10 ≡ 4 (mod 6), and 4^2 = 16 ≡ 4 (mod 6). So no matter what power of 4 (or 10) you have, it will always be equivalent to 4 (mod 6). Hence 10^100 ≡ 4 (mod 6).

Thus 10^(10^100) ≡ 10^4 (mod 7).
≡ 3^4 = 81 (mod 7)
≡ 4 (mod 7).

Answer 2

OK. Here's how I started.
I evaluated x Mod 7 for the numbers 10,000,000. Here's what I got
10 Mod 7 = 3
100 Mod 7 = 2
1000 Mod 7 = 6
10000 Mod 7 = 4
100000 Mod 7 = 5
1000000 Mod 7 = 1
10000000 Mod 7 = 3

and from there on it repeats.

Cutting and pasting over and over found that 10^100 Mod 7 = 4

I also noticed something. 10 Mod 7 * 100 Mod 7 = 1000 Mod 7

In fact, (x * y) Mod 7 = (x Mod 7)(y Mod 7) Mod 7

For example, the 100000 Mod 7 could be found by saying (1000 * 100) Mod 7 = (1000 Mod 7)(100 Mod 7) Mod 7
or (6 * 2) Mod 7 = 12 Mod 7 = 5

This is only a partial answer. Maybe this is enough to get you started. I'm working on the rest.

Edited. Here's some more
10^100 Mod 7 = 4
10^200 Mod 7 = 2
10^300 Mod 7 = 1
It's stepping backwards two spaces in the series.

Therefore 10^1000 Mod 7 = 10^(10^3) Mod 7 = 4
Also, 10(10^4) Mod 7 = 4

I'm just guessing here, but I think 10(10^100) Mod 7 = 4.

Although I'd be the first to tell you I can't prove it mathematically.

(I did this in Excel by creating a list of numbers 1-300 and pasting the pattern in next to it over and over.)

Answer 3

we know 10^6 mod 7 = 1

so we need to evaluate 10^100 mod 6 say k

then 10^k mod 7

we know 10^any number mod 6 = 4

so 10^100 mod 6 = 4

so 10^(10^100) mod 7 = 10^4 mod 7 = 10000 mod 7 = 4