1

Q2 (11 points) Let n € N be fixed modulus such that n > 2 Consider Zn {[0], _ [n = 1]} the set of integers modulo n_ (a) Recall: for m.n € Z (that ar...

Question

Q2 (11 points) Let n € N be fixed modulus such that n > 2 Consider Zn {[0], _ [n = 1]} the set of integers modulo n_ (a) Recall: for m.n € Z (that are not both 0), we define gcd(m, n) min(S) _ where S = {k€ N: k = mr + ny for some T.y € Z} Using this definition for gcd, prove the following proposition: Proposition HW4 If 3[6] € Zn such that [a] [6] = [1], then gcd(a,n) = 1 Clearly state your assumptions and carefully justify the steps of your proof using appropriate definitions and

Q2 (11 points) Let n € N be fixed modulus such that n > 2 Consider Zn {[0], _ [n = 1]} the set of integers modulo n_ (a) Recall: for m.n € Z (that are not both 0), we define gcd(m, n) min(S) _ where S = {k€ N: k = mr + ny for some T.y € Z} Using this definition for gcd, prove the following proposition: Proposition HW4 If 3[6] € Zn such that [a] [6] = [1], then gcd(a,n) = 1 Clearly state your assumptions and carefully justify the steps of your proof using appropriate definitions and brief explanations: (b) Write the converse of the implication in Proposition HW4 c) Does the converse of Proposition HW4 hold? If so, prove it: Ifnot, provide concrete counterexample and briefly - explain. Definition_ Let [a] € Zn- An element [b] € Za is called multiplicative inverse for [a] if [a] @ [b] = [1]: Does [5] € Zu have multiplicative inverse? If s0, what is *it? Briefly justify your answer: Does [3] € Zm have multiplicative inverse? If S0, what is *it? Briefly justify your answer Which elements of Z have multiplicative inverses? For those elements (if any) find their multiplicative "inverses. Justify your answers. (IV) Which elements of Z; have multiplicative inverses? For those elements (if any) find their multiplicative *inverses. Justify your answers: #For all parts of (d) please represent multiplicative inverses using their canonical representatives [0]



Answers

It seems likely that $\sqrt{n}$ is irrational whenever the natural number $n$ is not the square of another natural number. Although the method of Problem 13 may actually be used to treat any particular case, it is not clear in advance that it will always work, and a proof for the general case requires some extra information. A natural number $p$ is called a prime number if it is impossible to write $p=a b$ for natural numbers $a$ and $b$ unless one of these is $p,$ and the other $1 ;$ for convenience we also agree that 1 is not a prime number. The first few prime universe are 2 $3,5,7,11,13,17,19 .$ If $n>1$ is not a prime, then $n=a b,$ with $a$ and $b$ both $<n ;$ if either $a$ or $b$ is not a prime it can be factored similarly; continuing in this way proves that we can write $n$ as a product of primes. For example, $28=4 \cdot 7=2 \cdot 2 \cdot 7$ (a) Turn this argument into a rigorous proof by complete induction. (To be sure, any reasonable mathematician would accept the informal argument, but this is partly because it would be obvious to him how to state it rigorously.) A fundamental theorem about integers, which we will not prove here, states that this factorization is unique, except for the order of the factors. Thus, for example, 28 can never be written as a product of primes one of which is $3,$ nor can it be written in a way that involves 2 only once (now you should appreciate why 1 is not allowed as a prime). (b) Using this fact, prove that $\sqrt{n}$ is irrational unless $n=m^{2}$ for some natural number $m$ (c) Prove more generally that $\sqrt[k]{n}$ is irrational unless $n=m^{k}$ (d) No discussion of prime numbers should fail to allude to Euclid's beautiful proof that there are infinitely many of them. Prove that there cannot be only finitely many prime numbers $p_{1}, p_{2}, p_{3}, \ldots$ $p_{n}$ by considering $p_{1} \cdot p_{2} \cdot \ldots \cdot p_{n}+1$.


Similar Solved Questions

5 answers
Solve the following cquation for x=17Find f'(*) if f(x) - log,algebraic for: expression Write the following 2: 130 arcsee(6))
Solve the following cquation for x =17 Find f'(*) if f(x) - log, algebraic for: expression Write the following 2: 130 arcsee(6))...
5 answers
PROBLEM 3 Let W =+0 0,12 + I3 +IS =0subspace of RiFind busis of W and hence dedluce the ditnension W In other words; find veetors 01. U2 . slow that;vk aId(a) spn{01, %2.Vk }the sulset {31,"2.Ut}linearly inlepeudlent.
PROBLEM 3 Let W = +0 0,12 + I3 +IS =0 subspace of Ri Find busis of W and hence dedluce the ditnension W In other words; find veetors 01. U2 . slow that; vk aId (a) spn{01, %2. Vk } the sulset {31,"2. Ut} linearly inlepeudlent....
5 answers
For the reactionC,H4(g) - HO(g) ~CH,CH OH(g)AG' ~6.1kJ and JS? 125.7 JK at 314K and JwThe maximum amount of work that could be done by" this reaction when 1.70 moles of € Hy(g) react at standard conditions at this temperatureSubmit AnswerRebry Entire Groupmore group attempts remaining
For the reaction C,H4(g) - HO(g) ~CH,CH OH(g) AG' ~6.1kJ and JS? 125.7 JK at 314K and Jw The maximum amount of work that could be done by" this reaction when 1.70 moles of € Hy(g) react at standard conditions at this temperature Submit Answer Rebry Entire Group more group attempts re...
5 answers
()f = <iH Jo BJiE 347 SI 484M u34} "8 SI [ -Juy[ 34} punoie PllOS 341 Bu[oADI Aq pourenqo PIIOs 341 Jo Jun[oA 341 pue LV S1 S1E-I 941 pUnoje H Bu_A[OADI Aq paugenqo Plos 341 JO Ju[OA 341 JI mopq papeys H uorBa1 341 JopISUo) (SBuo4L s9#)
()f = < iH Jo BJiE 347 SI 484M u34} "8 SI [ - Juy[ 34} punoie PllOS 341 Bu[oADI Aq pourenqo PIIOs 341 Jo Jun[oA 341 pue LV S1 S1E-I 941 pUnoje H Bu_A[OADI Aq paugenqo Plos 341 JO Ju[OA 341 JI mopq papeys H uorBa1 341 JopISUo) (SBuo4L s9#)...
5 answers
(1Opt) Prove that if n and b are positive integers with b 2 2. the base represeutarion of n has [logt "J - digits
(1Opt) Prove that if n and b are positive integers with b 2 2. the base represeutarion of n has [logt "J - digits...
5 answers
TabtP: Data for Determining the Ka of an Acidic Cation Or Kb of a Basic Anion 0.010 M Salt pH [H;o'] [OH ] K,or K6 Solution NaCl 7.00 1.0*104-7 1.0*10^ NaE 157 2.7*104-8 3.7*10^ NaCN 10.70 20*10441 5.0"10LA NacIo 9.73 49 40u.10 53" 10^ 5 NaNO; 167 21*104-8 4.8'10^ NHCI 5,63 2.3*104-6 43*104 9 NHE 6.21 6.2440^ 1.6'10^ 3 NHCN 9.32 4.8*10^-10 21 10^ 5 CHNH'CN 729 5.1"10^-8 20'10~=
TabtP: Data for Determining the Ka of an Acidic Cation Or Kb of a Basic Anion 0.010 M Salt pH [H;o'] [OH ] K,or K6 Solution NaCl 7.00 1.0*104-7 1.0*10^ NaE 157 2.7*104-8 3.7*10^ NaCN 10.70 20*10441 5.0"10LA NacIo 9.73 49 40u.10 53" 10^ 5 NaNO; 167 21*104-8 4.8'10^ NHCI 5,63 2.3*...
5 answers
How many grams of chlorine gas are needed make10"solution that is 1.00 ppm chlorine by mass?gCl
How many grams of chlorine gas are needed make 10" solution that is 1.00 ppm chlorine by mass? gCl...
5 answers
Prelab: Determine the solution concentrations of each of the diluted solutions We will arbitrarily callthe stock solution as 1.OOM. The final volume for all five solutions is 250 mL: Remember the dilution equation: MVi MzVz: Record each concentration to three significant digits_SOLUTIONAMT: OF STOCK SOLUTION 5.00 mL 10.00 mL 20.00 mL 25.00 mL 30.00 mL
Prelab: Determine the solution concentrations of each of the diluted solutions We will arbitrarily callthe stock solution as 1.OOM. The final volume for all five solutions is 250 mL: Remember the dilution equation: MVi MzVz: Record each concentration to three significant digits_ SOLUTION AMT: OF STO...
5 answers
13 Let be finite abelian group of order n and let pk be the largest power of prime p dividing n. Show that ZlpkZ &2 A is isomorphic to the Sylow p-subgroup of A
13 Let be finite abelian group of order n and let pk be the largest power of prime p dividing n. Show that ZlpkZ &2 A is isomorphic to the Sylow p-subgroup of A...
5 answers
(a) Show that Approximation 1 of Section $6.8$ is equivalent to uniformizing the continuous-time Markov chain with a value $v$ such that ot $=n$ and then approximating $P_{U}(t)$ by $P_{U}^{* prime}$.(b) Explain why the preceding should make a good approximation.Hint: What is the standard deviation of a Poisson random variable with mean $n ?$
(a) Show that Approximation 1 of Section $6.8$ is equivalent to uniformizing the continuous-time Markov chain with a value $v$ such that ot $=n$ and then approximating $P_{U}(t)$ by $P_{U}^{* prime}$. (b) Explain why the preceding should make a good approximation. Hint: What is the standard deviatio...
1 answers
Find the area of the region that lies outside the first curve and inside the second curve. $r=4 \cos \theta, \quad r=2$
Find the area of the region that lies outside the first curve and inside the second curve. $r=4 \cos \theta, \quad r=2$...
1 answers
Evaluating a Limit at Infinity In Exercises $9 - 28$ , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { t \rightarrow \infty } \frac { t ^ { 2 } } { t + 3 } $$
Evaluating a Limit at Infinity In Exercises $9 - 28$ , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { t \rightarrow \infty } \frac { t ^ { 2 } } { t + 3 } $$...
5 answers
1. If an account earns 10% simple interest per year, how many years will it take $1 grow to $8?2. How much should you save in an account now so that you will have $1000 in 6 years, if the account earns 7% interest, compounded monthly?Two questions please, thank you so much! i really appreciate that!
1. If an account earns 10% simple interest per year, how many years will it take $1 grow to $8?2. How much should you save in an account now so that you will have $1000 in 6 years, if the account earns 7% interest, compounded monthly?Two questions please, thank you so much! i really appreciate that!...
5 answers
Find the solution of the initial value problem and describe thebehavior of the solution when t → ∞y'' − 4y' + 5y = 0y(0) = 1, y'(0) = 0
Find the solution of the initial value problem and describe the behavior of the solution when t → ∞ y'' − 4y' + 5y = 0 y(0) = 1, y'(0) = 0...
5 answers
25.Place the compounds AB, AD, BD, and AC in order of increasing ionic character? Difference in electronegativities are AB 0.5,AD 2.5,BD 2.0, and AC 1.0: Points)AD, BD; AC, ABAD, AC, AB, BDAB, AC, BD, ADAB, AD, BD; AC
25.Place the compounds AB, AD, BD, and AC in order of increasing ionic character? Difference in electronegativities are AB 0.5,AD 2.5,BD 2.0, and AC 1.0: Points) AD, BD; AC, AB AD, AC, AB, BD AB, AC, BD, AD AB, AD, BD; AC...
5 answers
5. Find the transpose of the augmented matrix of the linear system. (10 points)7x1 5x2 + 2x3 + X4 = 1 ~X1 + 2x2 + 4x3 ~ x4 = 5 3X1 7x2 +X3 + 9x4 =-6
5. Find the transpose of the augmented matrix of the linear system. (10 points) 7x1 5x2 + 2x3 + X4 = 1 ~X1 + 2x2 + 4x3 ~ x4 = 5 3X1 7x2 +X3 + 9x4 =-6...

-- 0.021480--