Harvard UniversityComputer Science 20In-Class Problems 7 - Number TheoryExecutive Summary•Divisibility: Ifn6= 0 andaare integers, we say thatndividesa(and writen|a) if there exists anmsuch thata=nm. Whenn|awe also saynis a divisor ofaandais a multiple ofn.•The Division Algorithm/Theorem: Ifaanddare integers anddis positive, then there areuniqueintegersqandrsuch thata=dq+rand 0≤r < d.•Euclidean GCD Algorithm:gcd (a , b :p o s i t i v ei n t e g e r s )x:=a ,y:=bwhile(y>0)r:=remainder (x , y)x:=yy:=rreturnxPROBLEM 1Prove the following claim directly from the definition of “divides” (i.e. don’t use facts about divides provedin class or the book). A direct proof should work. You should not have/need a fraction or a division sign inyour proof. Make sure all your variables are defined and quantified properly!Claim: For any integersp,q, andr, wherepis non-zero, ifp|3qand 3q|r, thenp|3q+r.Solution.Hints:•Ensure that students identify the givens and the assumptions they should make from the antecedentof the implication.•Encourage students to translate/unpack the divides operator into an algebraic expression by using thedefinition.•Ensure students are using the divides operator in the correct order. Read “4|12” as “4 divides 12”,which means 12 can be divided by 4 with no remainder.•Ensure students have introducedp,qandrcorrectly at the beginning of the proof (they might use“let...” or “for any...”. Their domain should be specified. The hypotheses then specify their relation-ships.•Ensure that students are not using facts about divisibility other than the definition.•Students should not have a fraction or a division sign anywhere in the proof.