Math 373 Homework 3
Due on Feb 11
1.Find the general solution (if solutions exist) of each of the following
linear Diophantine equations:
1. 2x+3y=4
gcd(2, 3) = 1 which divides 4. So the linear Diophantine equation
have integer solutions. In order to find
Math 373 - Test II Solutions
Thursday, 28 Oct 2010
Each problem is worth ten points. Please write clearly and neatly. These problems all require proofs, which must be
written in complete English sentences. For full credit please show all steps and give re
Math 373 Homework 2
Due on Feb 4
1.Use Mathematical induction to prove that for any positive integer n
we have
1 + 3 + 5 + + (2n 1) = n2 .
Proof. When n = 1, both sides of the identity are equal to 1.
Now suppose that the identity holds for n = k, namely
Math 373 Homework 6
Due on Mar 3
1.Find all solutions of each of the following system of congruences.
(a)
x1
x3
x5
(mod 3)
(mod 5)
(mod 7)
Solution.
c1 = 1, c2 = 3, c3 = 5
n1 = 35, n2 = 21, n3 = 15
n1 = 2, n2 = 1, n3 = 1
3 5 7 = 105
So by the Chinese Rema
Math 373 Homework 4
Due on Feb 18
1.Find all the integers x such that 9x 8 (mod 7).
Solution. Since 9 2 (mod 7) and 8 1 (mod 7), the congruence
equation can be restated as
2x 1
(mod 7).
Now observe that the reciprocal of 2 modulo 7 is 4, in that 2 4 =
8 1
Math 373 Homework 11
Due on April 21
1. Find all the quadratic residues for
(a) p = 11
Solution. We may check that 12 1 (mod 11), 22 4 (mod 11),
32 9 (mod 11), 42 5 (mod 11), 52 3 (mod 11) (no need to
go beyond 5). So the quadratic residues mod 11 are 1,
Math 373 Homework 10
Due on April 7
1. Find the prime factorization of 29!.
Solution.
b
29
29
29
29
c + b c + b c + b c = 14 + 7 + 3 + 1 = 25
2
4
8
16
29
29
29
b c + b c + b c = 9 + 3 + 1 = 13
3
9
27
29
29
b c+b c=5+1=6
5
25
29
b c=4
7
29
b c=2
11
29
b c=
Math 373 Homework 1
Due on Jan 28
1.Prove that
3 is an irrational number.
Proof. Suppose 3 is equal to the rational number p/q, with p, q not
sharing common divisor bigger than 1. Then 3q 2 = p2 , which means p
is divisible by 3. Let p = 3p1 . Then we hav
Math 373 Homework 7
Due on Mar 10
1.Recall d(n) is the divisor function. Then use the formula discussed
in class to compute
(a) d(72)
Solution.
72 = 23 32
so
d(72) = (3 + 1) (2 + 1) = 12
(b) d(2310)
Solution.
2310 = 2 3 5 7 11
so
d(2310) = 2 2 2 2 2 = 32
Math 373 Homework 5
Due on Feb 25
1.Find a
, the inverse of a modulo m, when
(a) a = 3 and m = 8
Solution. 3 3 1 (mod 8). a
= 3.
(b) a = 7 and m = 10
= 3.
Solution. 7 3 1 (mod 10). a
(c) a = 12 and m = 19.
Solution. 12 8 96 1 (mod 19). a
= 8.
Warning.
Math 373 Homework 9
Due on Mar 31
1. Find all the primitives roots for the modulus m.
(a) m = 13
Solution. (13) = 12. The only possible exponents are 1, 2, 3, 4,
6, 12. But 22 4 (mod 13), 23 8 (mod 13), 24 3 (mod 13)
and 26 12 (mod 13). So 2 must belong t
Math 373 Review sheet
Test II: Thurs, 28 Oct
Sun, 24 Oct 2010 / A Kumjian
Review: Tues, 26 Oct
Test I will cover sections 3.1 3.4, 4.1 4.5, 5.1 5.5, 6.1. You may bring a formula sheet to the test; it may include
denitions, results, propositions etc., but
Math 373 - Test I Solutions
Thursday, 23 Sept 2010
Each problem is worth ten points. Please write clearly and neatly. For full credit please show all steps and give reasons.
1. Describe the following set as cfw_x N : p(x), where p(x) is some property on x
Math 373 - Quiz 7 solutions
Thursday, 21 Oct 2010
1. Use the Intermediate Value Theorem to prove that there exists x (0, 2) such that
x3 2x2 + 3x = 4.
Let f (x) = x3 2x2 + 3x. Since f is a polynomial, it is continuous on [0, 2]. Now
f (0) = 0 < 4 < 6 = f
Math 373 - Quiz 3 solutions
Thursday, 16 Sept 2010
1. Consider the following open sentences P (x) and Q(x) over the domain S = R.
P (x) : |x| 3;
Q(x) : x 2.
(a) Determine the truth values of the following statements: P (0) Q(0); P (3) Q(3); P (7) Q(7).
Fo
Math 373 - Quiz 8 solutions
Thursday, 4 November 2010
1. Use mathematical induction to prove that 7n + 5 < 2n for every integer n 6.
Proof. Let P (n) denote this inequality (where n N). We use induction to prove: P (n) for every integer n 6.
Base step : F
Math 373 - Quiz 9 solutions
Tuesday, 16 Nov 2010
1. Let R be the relation dened on Z by a R b if a2 b2 (mod 4). Given that R is an equivalence relation, determine
the distinct equivalence classes. [Hint: For all a, b Z, if a b (mod 4) then a2 b2 (mod 4)].