1. Let n be an integer. Show that gcd (n², n² + n + 1) = 1. Note: You must justify every step of your proof using a result shown either in this course or in a previous one. Steps without a proper justification will not account for credit. 2. Express the following in base 10. Show all the necessary work to obtain your answer. (a) 12345 (b) 101012 (c) 11111 3. a) Convert the base 10 number 54321 to base 6. Show all the necessary work to obtain your answer. b) Convert the base 10 number 100 to base 2. Show all the necessary work to obtain your answer. 4. 6. For each of the following equations, find all integral solutions or show that it has none. Show all the necessary work to obtain your answer. (a) 3x+4y=10 (b) 44x-17y = 9 (c) 60x+9y= 31 (d) 16x + 24y = 44 5. What is the smallest nonzero value of X Y - where x and y are integers? Show all the necessary 136 31 work to obtain your answer. 6. Find the prime factorization of the following integers. Show all the necessary work to obtain your answer. (a) 13736 (b) 1728 7. Let a, b, c be integers. Show that if a + b,b+c, and a + c are multiples of 3, then each of a, b, c is a multiple of 3. Note: You must justify every step of your proof using a result shown either in this course or in a previous one. Steps without a proper justification will not account for credit. 8. Let p be a prime, and let a, b be nonzero integers. Suppose pc | c but pe+1 | a, and pd | b but pd+1 | b. a) Show that if c
1. Let n be an integer. Show that gcd (n², n² + n + 1) = 1. Note: You must justify every step of your proof using a result shown either in this course or in a previous one. Steps without a proper justification will not account for credit. 2. Express the following in base 10. Show all the necessary work to obtain your answer. (a) 12345 (b) 101012 (c) 11111 3. a) Convert the base 10 number 54321 to base 6. Show all the necessary work to obtain your answer. b) Convert the base 10 number 100 to base 2. Show all the necessary work to obtain your answer. 4. 6. For each of the following equations, find all integral solutions or show that it has none. Show all the necessary work to obtain your answer. (a) 3x+4y=10 (b) 44x-17y = 9 (c) 60x+9y= 31 (d) 16x + 24y = 44 5. What is the smallest nonzero value of X Y - where x and y are integers? Show all the necessary 136 31 work to obtain your answer. 6. Find the prime factorization of the following integers. Show all the necessary work to obtain your answer. (a) 13736 (b) 1728 7. Let a, b, c be integers. Show that if a + b,b+c, and a + c are multiples of 3, then each of a, b, c is a multiple of 3. Note: You must justify every step of your proof using a result shown either in this course or in a previous one. Steps without a proper justification will not account for credit. 8. Let p be a prime, and let a, b be nonzero integers. Suppose pc | c but pe+1 | a, and pd | b but pd+1 | b. a) Show that if c
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 52E
Related questions
Question
Transcribed Image Text:1. Let n be an integer. Show that gcd (n², n² + n + 1) = 1.
Note: You must justify every step of your proof using a result shown either in this
course or in a previous one. Steps without a proper justification will not account for
credit.
2. Express the following in base 10. Show all the necessary work to obtain your answer.
(a) 12345 (b) 101012 (c) 11111
3. a) Convert the base 10 number 54321 to base 6. Show all the necessary work to obtain your answer.
b) Convert the base 10 number 100 to base 2. Show all the necessary work to obtain your answer.
4. 6. For each of the following equations, find all integral solutions or show that it has none. Show all the
necessary work to obtain your answer.
(a) 3x+4y=10 (b) 44x-17y = 9 (c) 60x+9y= 31 (d) 16x + 24y = 44
5. What is the smallest nonzero value of
X
Y
-
where x and y are integers? Show all the necessary
136 31
work to obtain your answer.
6. Find the prime factorization of the following integers. Show all the necessary work to obtain your
answer.
(a) 13736 (b) 1728
7. Let a, b, c be integers. Show that if a + b,b+c, and a + c are multiples of 3, then each of a, b, c is a
multiple of 3.
Note: You must justify every step of your proof using a result shown either in this
course or in a previous one. Steps without a proper justification will not account for
credit.
Transcribed Image Text:8. Let p be a prime, and let a, b be nonzero integers. Suppose pc | c but pe+1 | a, and pd | b but pd+1 | b.
a) Show that if c<d, then pc | a+b but pc+1 a + b.
b) Give an example to show that if c=d, then it is possible that pc+1 | a + b.
Note: You must justify every step of your proof using a result shown either in this
course or in a previous one. Steps without a proper justification will not account for
credit.
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