2. (a) Prove or disprove that {-39, 72, —23, 50, —15, 63, -52} is a complete residue system modulo 7. (b) Find a complete residue system modulo 7 consisting entirely of even integers. (c) Find a complete residue system modulo 7 consisting entirely of odd integers.

the residue classes modulo 3 in Z + are: {1, 4, 7, 10, . . . } {2, 5, 8, 11, . . . } {3, 6, 9, 12, . . . } Write the residue classes modulo n in Z + for (a) n = 4 (b) n = 5

Construct a product table showing all the Possible two-digit numbers from the list ( 1,2,3,4,5,6) Of the thirty-lixe number in the porduct table you just made, bist the ones that belong to each of the following categories. (a) odd numbers (6) Numbers with repeating digits. (c) Multiples of 6 (a) Prime numbers 6) Triangular members () Square numbers. (g) Fibonace numbers (4) Power & 2

Supposed A = {n|2 < n < 16 and n is odd} and B= {n|n is prime and 2 < n < 11}. An integer is prime if it's only divisors are 1 and the integer by itself.

Prove by induction: Prove that for all natural numbers k and a that 3((5*2^(2k+1) -1)/3 + a*4^k+1) is in the form of (5+6a)* 2^j

Give a proof by contradiction that there does not exist an integer which is both even and odd.  Also, use the exact definitions of odd integers and even integers.