怎樣計呢...?

By Reminder Theorem, the required remainder = (-1)1000 + 6 = 7.
Since the results in (a) is true for all x, now we put x = 8.
Then the devidend = 81000 + 6 and the divisor = 8 + 1 = 9
Hence the required remainder = 7 again.
Remarks: Reminder Theorem
When a polynomial P (x) is divided by (x - a), then P (x) can be writen as a form

P (x) = Q (x)*(x - a) + R, where Q (x) is another polynomial which degree is smaller than P (x) and R is a constant.

This follows directly from the Division Algorithm, and hence the remainder R can be found by putting x = a into the identity, i.e. P (a) = Q (a)*(a - a) + R, which implies P (a) = R.

8^1000 + 6 (mod 9)
= (-1)^1000 + 6
= 1 + 6
= 7//

**a = c (mod b) --> remainder theorem
it means that c is the remainder when a is divided by b
e.g. 286 = 0 (mod 2)
156 = 1 (mod 5)

Consider f(x) = x^1000 + 6
When f(x) is divided by (x+1), the remainder is f(-1) = (-1)^1000 + 6 = 1+6 = 7
Put x = 8, when f(8) = 8^1000 + 6 is divied by 9 , the remainder is 7