Re: problem of arithmetic modulo

On Tue, 22 Jan 2008, Bill wrote:

We have 3^1000 = sum(a_i 10^i) where a_i are the digits of 3^1000
expressed in decimal system.

let 's say c=sum(a_i) the sum over all digits.
the value of c modulo 3 equals 0 and it's easily solved.

But what 's about c modulo 7 ?

Let dj = a_j.

3^1000 = sum_j dj.10^j = 3 * sum_j dj (mod 7)

3^6 = 1 (mod 7)
3^1000 = (3^6)^166 * 3^4 = 3 * 3^3 = 3 * 27 = -3 (mod 7)

sum_j dj = -1 = 6 (mod 7)
.

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