Question

I dont understand the steps to getting to the answer.

Answer 1

(a) Fermat's little theorem states that for any prime p , we have

a^p-1 = 1 (mod p).

for any integer ,a. Then

999999 = 1000000 - 1 = 10^6 - 1 = 1 - 1 = 0 (mod 7)

so 7 does indeed divide 999,999.

(b) Generalizing, we have

10^12n - 1 = 999...999 (12n nines) = (10^n)^12 - 1 = 0 (mod 13)

for positive integer, n. Now, since 1 = 1001 - 1000 and -1 = (mod 1001) , it follows that 10^3 is its own inverse modulo 1001, i.e.

10^3 x = 1 (mod 1001) -----> x = 10^3

This means

10^12n = (10^3 x 10^3)^2n = 1^2n = 1 (mod 1001) ----> 10^12n - 1 = 0 (mod 1001)