class7 numbertheory s.pdf - Harvard University Computer...

Harvard University
Computer Science 20
In-Class Problems 7 - Number Theory
Executive Summary
• Divisibility : If n 6 = 0 and a are integers, we say that n divides a (and write n | a ) if there exists an
m such that a = nm . When n | a we also say n is a divisor of a and a is a multiple of n .
• The Division Algorithm/Theorem : If a and d are integers and d is positive, then there are unique
integers q and r such that a = dq + r and 0 ≤ r < d .
• Euclidean GCD Algorithm :
gcd (a , b : p o s i t i v e i n t e g e r s )
x := a , y := b
while (y > 0)
r := remainder (x , y)
x := y
y := r
return x
PROBLEM 1
Prove the following claim directly from the definition of “divides” (i.e. don’t use facts about divides proved
in class or the book). A direct proof should work. You should not have/need a fraction or a division sign in
your proof. Make sure all your variables are defined and quantified properly!
Claim: For any integers p , q , and r , where p is non-zero, if p | 3 q and 3 q | r , then p | 3 q + r .
Solution.
Hints:
• Ensure that students identify the givens and the assumptions they should make from the antecedent
of the implication.
• Encourage students to translate/unpack the divides operator into an algebraic expression by using the
definition.
• 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 introduced p , q and r correctly 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.