Math 261 - Introduction to Number Theory Quiz#2 Solution Show all your work for full credit.
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1. (a) (4 points) State the definition of the least common multiple of two integers a and b. The least common multiple of two integers a and b is the smalle
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 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. Suppose a + d = n, where n is an integer. Then ad + bc = bd
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
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 defined by: aRb if ab 1. Is R an equival
Math 261 - Introduction to Number Theory Homework 5 Solution 1. Find all primes p such that 7p + 4 is a square, i.e. find 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). Si
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 Quiz#5 Solution Show all your work for full credit.
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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 Show all your work for full credit.
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1. (8 pts) Abraham Lincoln was shot at Ford's Theatre on the evening of April 14, 1865 and died the following morning. Determine on which day of the week Li
Math 261 - Introduction to Number Theory Quiz#3 Solution Show all your work for full credit.
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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#1 Solution Show all your work for full credit. 1. (4 pts) Compute the following numbers:
3
Name:
(i)
i=1
i! = 1! + 2! + 3! = 1 + 2 + 6 = 9.
43
(ii)
(i + 7) = 43 + 7 = 50.
i=43
2. (5 pts) State the well-orderin
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
1. (a) (8 pts) Compute the number
i=2 4 5 5 j=i
j .
5 5
j
i=2 j=i
=
j=2
j
j=3
j
j=4
j
= (2 + 3 + 4 + 5)(