1.4
J.A.Beachy
1
Review Problems
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
1. Prove that if a, b, c are integers for which b | a and b | (a c), then b | c.
Solution: Assume that b | a and b | (a c).
2.1
2.1
J.A.Beachy
1
Functions
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
21. The Vertical Line Test from calculus says that a curve in the xy-plane is the graph
of a function of x if and only if no
2.1
2.1
J.A.Beachy
1
Functions
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
Besides reading Section 2.1, it might help to get out your calculus textbook and review
composite functions, one-to-one and o
2.2
2.2
J.A.Beachy
1
Equivalence Relations
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
In a variety of situations it is useful to split a set up into subsets in which the elements
have some property i
2.2
2.2
J.A.Beachy
1
Equivalence Relations
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
13. For the function f : R R dened by f (x) = x2 , for all x R, describe the
equivalence relation f on R that is
1.4
1.4
J.A.Beachy
1
Integers Modulo n
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
31. Find the multiplicative inverse of each nonzero element of Z7 .
Solution: Since 6 1 (mod 7), the class [6]7 is it
1.4
1.4
J.A.Beachy
1
Integers Modulo n
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
The ideas in this section allow us to work with equations instead of congruences, provided
we think in terms of equiv
1.3
1.3
J.A.Beachy
1
Congruences
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
29. Solve the congruence
42x 12 (mod 90).
Comment: You need to recall Theorem 1.3.5, which states that ax b (mod n) has
a s
1.1
1.1
J.A.Beachy
1
Divisors
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
25. Let a be an integer. Show that if 2 | a and 3 | a, then 6 | a.
Solution: Suppose that 2 | a and 3 | a. Then we know from D
1.2
1.2
J.A.Beachy
1
Primes
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
26. (a) Use the Euclidean algorithm to nd gcd(1776, 1492).
Solution: We have 1776 = 1492 1 + 284;
284 = 72 3 + 68; 72 = 68 1 + 4
1.1
1.1
J.A.Beachy
1
Divisors
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
Before working through the solved problems for this section, you need to make sure that
you are familiar with all of the denit
1.3
1.3
J.A.Beachy
1
Congruences
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
In this section, it is important to remember that although working with congruences is
almost like working with equations,