## Question

###### 7 . Fix a positive m > 1 and define complete set of residues mod to be a set of integers S such that every integer is congruent to exactly one integer in the set $. For example, by theorem section 4 of Dudley; the set $ {0,1, mn = 1} is a complete set of residues mod m and is usually called the complete set of least residues mod m" Describe a complete set S of residues mod 13 where every I â‚¬ S satisfies 20 < = < 40. Describe a complete set S of residues mod 13 where every â‚¬ â‚¬

7 . Fix a positive m > 1 and define complete set of residues mod to be a set of integers S such that every integer is congruent to exactly one integer in the set $. For example, by theorem section 4 of Dudley; the set $ {0,1, mn = 1} is a complete set of residues mod m and is usually called the complete set of least residues mod m" Describe a complete set S of residues mod 13 where every I â‚¬ S satisfies 20 < = < 40. Describe a complete set S of residues mod 13 where every â‚¬ â‚¬ S is negative: Describe complete set S of residues mod 13 consisting entirely of odd integers Assume that m > 1 is an odd positive integer: Show m _1 m -3 S = {= m -3 "24} 1-1,0,1, is a complete set of residues mod Explicitly list this complete set of residues for mod 13. Let a > 1, (a,m) = 1 and S is a complete set of residues mod m. Form the new set Sa = {arlz â‚¬ S}. Prove Sa is complete set of residues mod m. Show by example that the result will fail if (a,m) >1