MATHE MATI CS 220—DISCRETE MATH EMAT ICS —S EPTEMBER 5, 2006
ASSI G NMENT 1
When the problems below ask you to “prove” something, they mean it! Please follow the examples
in the book:
•Label the statement to be proved “Theorem” and the proof “Proof.”
•Write in complete sentences.
We will work extensively on writing proofs during Math 220. Far more guidelines will be provided
as we develop both more mathematics and more methods of proof.
Please turn your homework in to the folder labeled “MATH 220” hanging on the wall outside the
Mathematics Department office, King 205.
When you turn in your homework, please be sure to remove any spiral-notebook scritchies and, if
necessary, to staple multiple skeets together. The Mathematics Department office has both a stapler
and a paper cutter available for student use.
Note: Biggs has included answers (although not full solutions) to essentially all exercises in the first
nine chapters of our textbook. Thus, while the problems in the book provide wonderful opportunities
for self-study, the assignments you turn in will not be taken directly from the book.
ASSI GN ME NT 1 : DU E FRI DAY, SEPT EMB ER 8 .
Reading. Sections 1.1–1.3 of Biggs, with particular attention to the “exercises” in section 1.2.
Problems.
(1) Determine whether each of the following statements is true or false. Then, write up a proof
of the statement or its negation, as appropriate.
(a) •91 is not prime and 93 is not prime.
(b) •293 is a perfect cube or 231 −1is prime.
(c) •If 17 is a multiple of 7, then 51 is a multiple of 21.
(2) Define a pair of numbers, (n, m), to be compatible if nand mhave a common factor greater
than the square root of the smaller of mand n.
(a) Prove the statement
•The pair (12,18) is compatible.
(b) Find a non-compatible pair of numbers, both greater than 100. Prove that the pair is
not compatible.
(c) Find an infinite list of pairs of compatible numbers. (You don’t need to give a proof
that every pair on your list is actually compatible. However, you should give enough
pairs that the pattern for keep generating pairs is clear—of course, giving formulas
would be even better.)