Question

use the fact that 10 is congruent to -1(mod 11) to devise a test for divisibility by 11 thats similar to our tests for divisibility by 3 and 9

Answer 1

for a number n to be divisible by 11, n mod 11 must be congruent to 0.
now express n in base 10 form as ...fedcba, or a+10b+100c+1000e+10000f+...
calculate n mod 11, using the fact that 10 is congruent to -1 mod 11:
a+10b+100c+... mod 11
=a-b+c-d+... mod 11
since 10^n mod 11=(10 mod 11)^n
and a*b mod 11=a*(b mod 11)
thus, if a-b+c-d+... is a multiple of 11, the number is divisible by 11.
so for example for 649, 9-4+6=11, thus 649 is divisible by 11.