Math 261 Introduction to Number Theory
Exam #2 Solution
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There are a total of 5 questions in this exam. Show all your work for full credit.
1. (16 pts) Let R be a binary relation on the set of all integers Z dened by: aRb if ab 1. Is
R an equivalence
MA 261
Worksheet 1/29
Name:
More Induction Practice
Theorem A.20: For every natural number n > 3, 2n < n!.
Proof:
Denition: The bonacci numbers are a sequence of numbers, denoted fn , dened as follows:
f1 = 1,
f2 = 1,
and
fn = fn1 + fn2
for
n3
Exercise 2a
MA 261
Worksheet 1/27
Name:
Exercise 1.17 Let a, b, k, and n be integers with n > 0 and k > 1. Show that if a b (mod n) and
ak1 bk1 (mod n) then ak bk (mod n). (Feel free to make use of Theorem 1.14; you will prove it
in Homework 2.)
Proof by Induction: T
MA 261
Worksheet 1/25
Name:
Recall from the reading that we have the following denition:
Denition: Suppose that a, b, and n are integers, with n > 0. We say that a and b are congruent
modulo n if and only if n|(a b) and we denote it by a b (mod n).
Use th
MA 261
Worksheet 2/1 B
Name:
The Division Algorithm
The Division Algorithm: Let n and m be natural numbers. Then there exist unique (!) integers q
and r such that m = nq + r and 0 r < n.
Now that youve had a bit of time to think about the proofs, Ill give
MA 261
Worksheet 1/20
Name:
Recall from the reading that we have the following denition:
Denition: If d and a are integers then d divides a, written d|a, if and only if there is an integer k
such that a = kd.
Use this denition to prove the following theor
MA 261
Worksheet 2/1
Name:
The Division Algorithm
The Division Algorithm: Let n and m be natural numbers. Then there exist unique (!) integers q
and r such that m = nq + r and 0 r < n.
Note: When we say q and r are unique we mean that if we could nd anoth
MA 261
Worksheet 2/5
Name:
Prove the following theorems:
Theorem 1.32 Let a, n, b, r, and k be integers. If a = nb + r and k|a and k|b then k|r.
Theorem 1.33 Let a, b, n1 , and r1 be integers with a and b not both 0. If a = n1 b + r1 then gcd(a, b)
= gcd(
MA 261
Worksheet 2/8
Name:
Exercise A: Use the Euclidean Algorithm to show that 162 and 31 are relatively prime. (i.e.(162, 31) = 1)
Exercise B: Use the Euclidean Algorithm backwards to nd integers x and y such that 162x + 31y = 1.
Theorem 1.38 If a, b Z
MA 261
Worksheet 2/17
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Exercise 0: Circle each of the following equations that have an integer solution:
4x + 10y = 8
15x + 5y = 7
Exercise 1: Find all integer solutions to 210x + 45y = 30.
12x + 6y = 3
6x + 3y = 9
Exercise 2: Prove that if x = x0 ,
MA 261
Worksheet 2/24
Name:
Exercise: Circle all the numbers between 1 and 100 that are prime and cross out the ones that are not
prime. What are some good strategies to use to make this quicker? Can you determine a systematic way
to do this for longer li
MA 261
Worksheet 2/26
Name:
Theorem 2.7 (Fundamental Theorem of Arithmetic Existence) For every n N, n > 1, n is either
prime or it can be expressed as a nite product of prime numbers.
(Hint: Use strong induction! It will be vaguely similar to the homewor
MA 261
Worksheet 3/2
Exercise: Prove that if p is prime then
Name:
p is irrational.
The following theorem is equivalent to Theorem 1.42 which you already proved using linear combinations
and substitution. Now prove it using the Fundamental Theorem of Arit
Implications
An implication is a logical statement of the form "if P , then Q." Symbolically we write this as
P
Q, there are three related statements:
For any given logical statement P
The converse, Q
P
The contrapositive, Q
The inverse, P
Q.
P
Q.
MA 261
Worksheet 1/15
Name:
To prove a statement of the form if P then Q you must rst assume P and then show Q. Use the
denitions of even, odd, and rational to prove the following statements.
1. If x is even and y is odd then x + y is odd.
2. If x is even
Optional Homework: Pythagorean Triples
The Pythagorean triples are a set of three integers (a, b, c) which satisfies a2+b2=c2. The
Pythagorean theorem is derived from that definition. In other words, the theorem states that the
sum of the squares of the r
Math 261 Introduction to Number Theory
Exam #1 Solution
Name:
There are a total of 5 questions in this exam. Show all your work for full credit.
4
5
i=2
j =i
1. (a) (8 pts) Compute the number
4
5
i=2
j.
j =i
5
j
=
5
j
5
j
j =2
j =3
j
j =4
= (2 + 3 + 4 + 5
Math 261 Introduction to Number Theory
Quiz#5 Solution
Name:
Show all your work for full credit.
1. (a) (6 pts) Euler found a generalization of Fermats Little Theorem. What does his result
say?
Solution : Eulers Theorem says if gcd(a, m) = 1, then a(m) 1
Math 261 Introduction to Number Theory
Quiz#4 Solution
Name:
Show all your work for full credit.
1. (8 pts) Abraham Lincoln was shot at Fords Theatre on the evening of April 14, 1865 and died
the following morning. Determine on which day of the week Linco
Math 261 Introduction to Number Theory
Quiz#3 Solution
Name:
Show all your work for full credit.
1. (a) (4 pts) What does the Fundamental Theorem of Arithmetic say?
Solution : Every integer n > 1 is either prime or a product of primes, and the product is
Math 261 Introduction to Number Theory
Quiz#2 Solution
Name:
Show all your work for full credit.
1. (a) (4 points) State the denition of the least common multiple of two integers a and b.
The least common multiple of two integers a and b is the smallest p
Math 261 Introduction to Number Theory
Quiz#1 Solution
Name:
Show all your work for full credit.
1. (4 pts) Compute the following numbers:
3
(i)
i! = 1! + 2! + 3! = 1 + 2 + 6 = 9.
i=1
43
(ii)
(i + 7) = 43 + 7 = 50.
i=43
2. (5 pts) State the well-ordering
Math 261 Introduction to Number Theory
Homework 8 Solution
1. Use Eulers Theorem to show that n12 1 (mod 72) if gcd(n, 72) = 1.
Proof. Since 72 = 8 9, it suces to show that n12 1 modulo 8 and 9. Note that gcd(n, 72) =
1 implies that gcd(n, 8) = gcd(n, 9)
Math 261 Introduction to Number Theory
Homework 7 Solution
1. In class, we proved the following result: If a b (mod m), a b (mod n) and gcd(m, n) = 1,
then a b (mod mn).
Give an example to illustrate that this result is not necessarily valid if m and n ar
Math 261 Introduction to Number Theory
Homework 6
1. Show that no integer of the form 4k + 3 is the sum of two squares.
(Hint: For every integer n, what can n modulo 4 be? How about n2 modulo 4?)
Solution : For any integer n, n 0, 1, 2 or 3 (mod 4). So n2
Math 261 Introduction to Number Theory
Homework 5 Solution
1. Find all primes p such that 7p + 4 is a square, i.e. nd all primes p such that 7p + 4 = x2 for
some integer x.
Solution : The equation 7p + 4 = x2 can be rewritten as 7p = (x + 2)(x 2). Since 7
Math 261 Introduction to Number Theory
Homework 4 Solution
1. Given three integers a, b, c with c > 0, prove that lcm(ac, bc) = c lcm(a, b).
Solution :
ac bc
lcm(ac, bc) =
gcd(ac, bc)
abc2
=
c gcd(a, b)
ab
=c
gcd(a, b)
= c lcm(a, b).
2. Use the Euclidean
Math 261 Introduction to Number Theory
Homework 3 Solution
1. Let a, b, c and d be positive integers, with b = d. If gcd(a, b) = gcd(c, d) = 1, show that
cannot be an integer.
a
b
c
+d
c
Proof. Suppose a + d = n, where n is an integer. Then ad + bc = bdn.
Math 261 Introduction to Number Theory
Homework 2 Solution
1. Given a0 = 1, a1 = 7 and ak+2 4ak+1 + 3ak = 0 for all k 0, prove that an = 3n+1 2 for
all natural numbers n.
Proof. We will use the second form of weak induction. So for the base case, we need
MA 261
Worksheet 2/29
Name:
Exercise: Suppose a = 25 32 52 132 , b = 26 33 72 114 134 , c = 27 3 53 72 114 .
Does a|b? Does b|c?
Determine gcd(a, c) and gcd(b, c).
Determine lcm(a, c) and lcm(b, c).
Exercise: If a = pr1 pr2 prn and b = ps1 ps2 psn wher