Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 339 — #347
9.13. References 339
Problems for Section 9.3
Homework Problems
Problem 9.22.
TBA: Chebyshvev lower bound in prime density, based on Shoup pp.75–76
Problems for Section 9.4
Practice Problems
Problem 9.23.
Let p be a prime number and a 1 ; : : : ; a n integers. Prove the following Lemma by
induction:
Lemma.
If p divides a product a 1 a 2 a n ; then p divides some a i : (*)
You may assume the case for n D 2 which was given by Lemma 9.4.2.
Be sure to clearly state and label your Induction Hypothesis, Base case(s), and
Induction step.
Class Problems
Problem 9.24. (a) Let m D 2 9 5 24 11 7 17 12 and n D 2 3 7 22 11 211 13 1 17 9 19 2 . What
is the gcd.m; n/? What is the least common multiple lcm.m; n/ of m and n? Verify
that
gcd.m; n/ lcm.m; n/ D mn: (9.20)
(b) Describe in general how to find the gcd.m; n/ and lcm.m; n/ from the prime
factorizations of m and n. Conclude that equation (9.20) holds for all positive
integers m; n.
Homework Problems
Problem 9.25.
The set of complex numbers that are equal to m C n p 5 for some integers m; n
is called ZΠp 5. It will turn out that in ZΠp 5, not all numbers have unique
factorizations.