Download Test 2: Number Theory Review divides

Test 2: Number Theory Review
Part I: Definitions (3 points each * 8)
1. We say that a divides b, and write a | b if b = aq for some integer q.
2. We say that a is congruent to b mod n, and write a ! b (mod n) if a = b + nq for some
integer q.
3.
If a and b are integers, then the greatest common divisor of a and b equals d, and we
write (a,b) = d if
a. d >0
b. d | a and d | b
c. If c | a and c | b then c d
4. An integer p is prime if it has exactly two positive divisors.
5. The integer a is a unit mod n if there exists an integer b satisfying
(mod n).
6. If a is a unit mod n, then the order of a equals m, and we write o(a) = m if
a. m>0
b.
(mod n)
c. If t>0 and
(mod n) then
7. The well-ordering-principle (WOP) states that if S is a non-empty set of positive
integers then S has a least element.
8. The principle of weak induction states that if a set S of positive integers satisfies:
a.
b. If
then
then S = N, ie. S contains every positive integer.
9. The principle of strong induction states that if a set S of positive integers satisfies:
a.
b. If
for all k<t then
then S = N, ie. S contains every positive integer.
10. We say that a is less than b, where a and b are integers, and write a<b if there is some
natural number n satisfying a+b = b.
11. The axiom of trichotomy states that if a is an integer, then exactly one of the following is
true:
1. a< 0
2. a = 0
3. a>0
12. The associative property of addition states that if a,b,c are any integers then
(a+b)+c = a+(b+c)
13. The associative property of multiplication state that if a,b,c are any integers, then
(ab)c=a(bc)
14. The additive identity property states that if a is any integer, then a+0=0+a=a.
15. The mutliplicative identiy property states that if a is any integer, then a !1 = 1! a = a
16. The distributive property states that if a,b,c are any integers, then a(b+c)=ab+ac.
17. The additive inverse property states that if a is any integer, then there is an integer !a
satisfying a + (!a) = (!a) + a = 0
Part II: Computations, 50 points total. (10 points each)
1.
2.
3.
4.
5.
6.
7.
Convert numbers into different bases.
Perform operations in different bases.
Perform operations in different mods.
Find the order of an element in a mod.
Find the units in a mod.
Find the gcd of two integers.
Find the perfect squares.
Part 3: Proofs and other Problems 30 points total. (10 points each)
1.
2.
3.
4.
5.
6.
7.
8.
Proofs about logic and truth tables.
Proofs about divisibility.
Proofs about congruences.
Proof using induction or well-ordering.
Prove that
if n>2.
Proofs about divisibility
Proofs about mods
Proofs using induction or well-ordering.