Question

For each positive integer n, the set of integers {0,1,... ,n-1\} is known as the residue system modulo n. Within the residue system modulo 2^4, let A be the sum of all invertible integers modulo 2^4 and let B be the sum all of non-invertible integers modulo 2^4. What is A-B?

Answer 1

You'll have to go through each element in the set {0, 1, 2, ..., 15, 16} and see if it has a multiplicative inverse

For it to have an inverse, it would need to pair up with another number in that set to multiply to 1

For example, the inverse of 3 is 11 since
3*11 = 33 = 1 (mod 16)

another example: the inverse of 5 is 13 since
5*13 = 65 = 1 (mod 16)

Answer 2

A number like 2 does not have an inverse because there are no integral solutions to 2x = 1 (mod 16)

Answer 3

The answer is 8 thanks so much! @jim_thompson5910

Answer 4

All the ways to multiply to 1 in mod 16

1*1 = 1 (mod 16)
3*11 = 33 = 1 (mod 16)
5*13 = 65 = 1 (mod 16)
7*7 = 49 = 1 (mod 16)
9*9 = 81 = 1 (mod 16)
15*15 = 225 = 1 (mod 16)

So set A = {1, 3, 5, 7, 9, 11, 13, 15}. Basically all the odd numbers
Set B would then be B = {0, 2, 4, 6, 8, 10, 12, 14}

sum of A = 1+3+5+7+9+11+13+15 = 64
sum of B = 0+2+4+6+8+10+12+14 = 56

(sum of A) - (sum of B) = 64 - 56 = 8

So yes you are correct