Math 566 WI 2011 Midterm 1 Review and Practice Problems ...

Practice ProblemsRelations1. Prove whether or not each of the following relations is reflexive, symmetric, or transitive.If any relation is an equivalence relation, determine the equivalence classes.(a) The relation D defined on Z by m D n ⇔ m | 2n(b) The relation R defined on R by x R y ⇔ xy > 0(c) The set A = {−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5} with relation S defined byx S y ⇔ 3 | (x 2 − y 2 )Modular Arithmetic1. Let x be an odd integer. Prove that 16 | x 4 − 1.2. Let p be an odd prime such that p is the sum of two squares (that is, there existx, y ∈ Z such that p = x 2 + y 2 ). Prove that p ≡ 1 (mod 4).(It is also true that any prime congruent to 1 (mod 4) is a sum of two squares, butthat is much harder to prove).3. Take as given that 101 is invertible mod 1116. Assume that while you were doingsuccessive squaring to compute 101 331 mod 1116, you notice that(a) Prove that 101 30 ≡ 1 (mod 1116).101 2 ≡ 101 32 (mod 1116).(b) Compute 101 331 mod 1116 by using part (a).4. Given any integer a, the ordinary way to write a is as a sequence of digits a =a n a n−1 a n−2 . . . a 1 a 0 , where a i ∈ {0, 1, 2, . . . , 8, 9} for each 0 ≤ i ≤ n. By the sumof the digits of a, we mean the sum a n + a n−1 + · · · + a 1 + a 0 .(a) Prove that if the sum of the digits of a is divisible by 3, then a is divisible by 3.(b) Prove that if the sum of the digits of a is divisible by 3 and a is even, then a isdivisible by 6.Consecutive Integers1. How many integers from 1337 to 3614 inclusive are divisible by 4?2. How many integers from 1337 to 3614 inclusive are divisible by 3?3. How many integers from 1337 to 3614 inclusive are divisible by 4 and divisible by 3?