MATH 115 Final: MATH 115 UPenn 115FinalAns2012A
Document Summary
Remember that writing and correct use of notation are very important. Name: prove that if a, b and c are integers and a | b and b|c, then a | c. Proof: that if a, b and c are integers and a | b and b|c. Since a|b, then there exists an integer k such that b ka. Since b|c, then there exists an integer m such that c mb. Thus we see that c mb m ka mk a. Prove one of these facts (whichever one you want to). Both of these facts can be used throughout this exam if needed. Proofs: given in class or done in homework: prove that if a is any integer, then a2 2 is not divisible by 3. Then either a 0 (mod 3) or a 1 (mod 3), or a 2 (mod 3). If a 0 (mod 3), then a2 02 0 (mod 3)
