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 al
MA 261
Worksheet 1/29
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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
MA 261
Worksheet 1/27
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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;
MA 261
Worksheet 1/25
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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
MA 261
Worksheet 2/1 B
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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 youv
MA 261
Worksheet 1/20
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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
MA 261
Worksheet 2/1
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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 s
MA 261
Worksheet 2/5
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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 a
MA 261
Worksheet 2/8
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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
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 +
MA 261
Worksheet 2/24
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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
MA 261
Worksheet 2/26
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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 stro
MA 261
Worksheet 3/2
Exercise: Prove that if p is prime then
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p is irrational.
The following theorem is equivalent to Theorem 1.42 which you already proved using linear combinations
and substitut
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,
MA 261
Worksheet 1/15
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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
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,
Math 261 Introduction to Number Theory
Exam #1 Solution
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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
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?
Soluti
Math 261 Introduction to Number Theory
Quiz#4 Solution
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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 foll
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 e
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 co
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)
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
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
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?)
Solu
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
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,
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. Suppos
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
MA 261
Worksheet 2/29
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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)