Question 1171543: 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?
Please help me. I have tried multiple ways but have found nothing. Thank you
Answer by ikleyn(52754)   (Show Source):
You can put this solution on YOUR website! .
By modulo 2^4 = 16, invertible are those and only those that are mutually prime with 2.
They are 1, 3, 5, 7, 9, 11, 13, 15 (all odd numbers from 1 to 15, inclusive).
The rest, 0, 2, 4, 6, 8, 10, 12, 14 are not invertible.
So they want you calculate the difference
(1+3+5+7+9+11+13+15) - (0+2+4+6+8+10+12+14).
Regrouping, you see that this difference is 8*1 = 8. ANSWER
Solved.
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