# Monday, February 3rd, 2020

For Monday:

• Quizzes:  I somehow completely forgot to give the badges quiz on Friday, for which I apologize (I really try to avoid surprises).  We will take it on Monday instead.  That will also push the proof quiz back to Wednesday next week.
• Today we did a warmup direct proof and I left you with two proofs to write at home:
• Let a,b,c be integers.  Then $gcd(ca,cb) \ge c \gcd(a,b)$.
• If a,b,c are integers such that a|b and a|c then a|(b+c) and a|(b-c).
• In class we used the fact that the equation xy=1 has only two solutions in the integers, namely x=y=1 and x=y=-1.  Can you prove this?  Try contradiction!
• Review the proofs by contradiction from class and also the first Proposition in Chapter 6 of Hammack (first page of the chapter), which is an example.  Pay careful attention to the negation of the theorem, i.e. the supposition you make for contradiction.
• With your remaining time, try a few at home: Chapter 6 exercises 1,2,6,7,8,9 as time allows.