(1) Prove that is a, b, and c are integers with c + 0 then ac = bc = a = b ( that is this a key step towards showing that fractions work as expected bec makes dividing by c looks possible for any c + 0.)

Prove or disprove: For all integers a and b, if a +b is odd, then a is even and b is odd.

Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

Combinatorics Let i, j, k, n be positive integers. Suppose that a total of i + 2j + 2k people need to pay a bill for n dollars. Of these people: i are willing to pay any positive integer amount of dollars, j are only willing to pay 1 or 2 dollars, another j are willing to pay any positive odd integer amount of dollars, k are willing to pay 2 or 3 dollars, and another k are willing to pay any positive even integer amount of dollars. Find with proof the number of ways to split the bill as a function of i, j, k, and n.

Let a, b, c be positive natural numbers. Determine whether the following statement is true or false: If u > x and v > y then gcd(u, v) > gcd(x, y). True O False