Can anyone please help me out with these mathematical questions!? I really appreciate it!!!

Problem 5. Let A = N and let R be the relation: R = {(a,b) E N X N : ab is even} Is R reexive? symmetric? transitive? Is R an equivalence relation? If R is an equivalence relation describe, as precisely and concisely as you can, the equivalence classes of R and the corresponding partition of A. Problem 6. Let A = R2 \\ {(0,0} and dene a relation on A by: (3:1,y1)R(9:2,y2) if and only if there exists A E R, A 7A 0, such that ($2ay2) = A ' (3313311)- ls R reexive? symmetric? transitive? ls R an equivalence relation? If R is an equivalence relation describe as precisely and concisely as you can, the equivalence classes of R and the corresponding partition of A. Problem 7. We showed in class that given a map f: X > R we can dene an equivalence relation on X by: 1131721132 4:} f(131)= f($2). Describe, as precisely and concisely as you can, the equivalence classes and the partition of X for the relations dened by the following func- tions: a) f:lR2>lR,f(3:,y)=3:c|2y. b) fiR2>IR,f(93,y)=wy C) fiR2>IR,f(93,y)=4iv2 9y2- Problem 8. How many different equivalence relations are there on the set A = {1,2,3}? Problem 1. a) Find the inverse of 11 in Z123. b) Find all solutions to the equation 113: l 3 E 85 (mod 123). Problem 2. Find all solutions to the system of equations: 33: + 23; 4 (mod 11) :I: + 5y 7 (mod 11) Problem 3. a) Compute gcd(12!,1010). b) Compute gcd(3512,10024). c) What is the last digit of 20172017? (Hint: c1>(10) = 4) d) What are the last two digits of 20172017? (Hint: (100) : 40) e) What is the remainder when 7148 is divided by 39. (Hint: @(39) = 24) Problem 4. Let p E N be a prime number. We dened in class a function VpZN>Z20=NU{0} by the condition yp(a) = k if and only if pk divides a but p divide a. a) Prove that if a, b E N then yp(ab) = Vp(a) + 1110(1)). b) Let 02,!) E N. Guess and prove a formula for yp(a + b) in terms of Vp(a) and up(b). G) Let a E N prove that there exists I) E N such that a = b2 if and only if 149(0) E 0 (mod 2) for all prime numbers p. (Note: If a E N and a = b2, 5 E N, we say that a is a perfect square) (1) Let a E N prove that there exists 7' E Q such that a = r2 if and only if a is a perfect square. \"1 does not