Remainder theorem Definition and 63 Threads

Consider a certain integer between ## 1 ## and ## 1200 ##. Then ## x\equiv 1\pmod {9}, x\equiv 10\pmod {11} ## and ## x\equiv 0\pmod {13} ##. Applying the Chinese Remainder Theorem produces: ## n=9\cdot 11\cdot 13=1287 ##. This means ## N_{1}=\frac{1287}{9}=143, N_{2}=\frac{1287}{11}=117 ## and...

My first approach; ##(a-b)^3+(b-c)^3+(c-a)^3=a^3-3a^2b+3ab^2-b^3+b^3-3b^2c+3bc^2-c^3+c^3-3c^2a+3ca^2-a^3## ##=-3a^2b+3ab^2-3b^2c+3bc^2-3c^2a+3ca^2## what i did next was to add and subtract ##3abc## ...just by checking the terms ( I did not use...

##0=1+a+b+c## ##20=8+4a+2b+c## it follows that, ##13=3a+b## and, ##0=k^3+ak^2+bk+c##...1 ##0=(k+1)^3+a(k+1)^2+(k+1)b+c##...2 subtracting 1 and 2, ##3k^2+k(3+2a)+14-2a=0##

Dear How to solve the CRT for cryptography as below - (1) Find x such that x = 2(mod3) x = 5(mod9) x = 7(mod11) (2) Find x such that x = 2(mod3) x = 4(mod7) x = 5(mod11) (3) Find x such that x^2 = 26(mod77) (4) Find x such that x^2 = 38(mod77) Please help me by provide your advice and...

Homework Statement Let ##a, b, m, n## be integers with ##\gcd(m,n) = 1##. Let $$c \equiv (b-a)\cdot m^{-1} (\operatorname{mod} n)$$ Prove that ##x = a + cn## is a solution to ##x \equiv a (\operatorname{mod} m)## and ##x \equiv b (\operatorname{mod} n)##, (2.24). and that every solution to...

Let $m$ and $m'$ be positive integers, and $d=gcm(m,m')$. (i) The system: $x \equiv b (mod \ m)$ $x \equiv b' (mod \ m')$ has a solution if and only if $b \equiv b' (mod \ d)$ (ii) two solutions of the system are congruent $mod \ l$, where $l = lcm(m,m')$. I can prove part (i), but can...

Homework Statement What is the remainder when -3x^3 + 5x - 2 is divided by x? The Attempt at a Solution Not sure how to complete this one, I would assume that it is the same as x+0? How would you divide the last term, (-2). Please show your steps as this will help me a lot! Thanks!

Homework Statement (y4 - 5y2 + 2y - 15) / (3y - √(2)) The answer says (2√(2)/3)-(1301/81)...

NO TEMPLATE BECAUSE MOVED FROM ANOTHER FORUM Hello, I've been trying to figure out how it works for complicated problems, I know how to use long division, but I'm not understanding how this process is done for a problem like I have. Instructions: Write the function in the form ƒ(x) = (x -...

Homework Statement A polynomial P(x) is divided by (x-1), and gives a remainder of 1. When P(x) is divided by (x+1), it gives a remainder of 3. Find the remainder when P(x) is divided by (x^2 - 1) Homework Equations Remainder theorem The Attempt at a Solution I know that P(x) = (x-1)A(x) +...

Homework Statement [/B] Polynomial f(x) is divisible by ##x^2-1##. If f(x) is divided by ##x^3-x##, then the remainder is... A. ##(x^2-x)f(-1)## B. ##(x-x^2)f(-1)## C. ##(x^2-1)f(0)## D. ##(1-x^2)f(0)## E. ##(x^2+x)f(1)## Homework Equations Remainder theorem The Attempt at a Solution [/B]...

How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$? I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from...

Given any integer A, and a positive integer B, there exist unique integers Q and R such that $$A= B * Q + R$$ where $$ 0 ≤ R < B$$. When is says that $$Q$$ and $$R$$ are unique, what does that mean? That they are different from each other?

For any int $$n $$ , prove that $$ 4 | n (n^2 - 1) (n + 2)$$. I know I have to use the quotient remainder theorem, but I'm wondering how to go about this problem. I'm not sure how to phrase this problem in English.

Find the set of solutions $x=x(r,s,t)$ such that $(r+2\mathbb{N})\cap (s+3\mathbb{N})\cap (t+5\mathbb{N})=x+n\mathbb{N}.$ Hello MHB :). Any hints for the problem?

I have a few questions about the remainder theorem. 1: For series that "skip" terms (example: 1+x^2+x^4+x^6) the theorem says the n+1 derivative and x^(n+1)/(n+1)!. For example if you have 1 + x^2 where you know the next term would be x^4 you could treat it as a third order or a...

Homework Statement At what points ##x## in the interval ##(-1,1]## can one use the Lagrange Remainder Theorem to verify the expansion ##ln(1+x)=\sum_{k=1}^{\infty} (-1)^{k+1}{\frac{x^k}{k!}}##Homework Equations The Attempt at a Solution Now I know that ##ln(1+x)=\sum_{k=1}^{\infty}...

Q2.) Show all working out. a) Find the remainder when $$x^3+2x^2-5x-3$$ is divided by $$x-2$$. b) Find the remainder when $$x^3-3x^2-x+3$$ is divided by $$x-3$$.

Q1.) Use the factor and remainder theorems to find solutions to: $$1x^3+1x^2+-9x+D=0$$

Homework Statement Find each remainder: a. (x^3 + 5x^2 - 7x + 1) ÷ (x+2)(x-1)b. (2x^3 + x^2 - 4x - 2) ÷ (x^2 + 4x + 3)Homework Equations N/A. (We've used Long Division and Synthetic Division for previous questions.) The Attempt at a Solution How would i go about solving these? I'm pretty stuck.

Homework Statement When a polynomial is divided by (x+2), the remainder is -19. When the same polynomial is divided by (x-1), the remainder is 2. Determine the remainder when the polynomial is divided by (x-1)(x+2)Homework Equations The Attempt at a Solution had the polynomial been a real...

What is the most tangible way to introduce the Chinese Remainder Theorem? What are the practical and really interesting examples of this theorem. I am looking for examples which have a real impact on students.

Chinese Remainder Theorem What is the most tangible way to introduce the Chinese Remainder Theorem? What are the practical and really interesting examples of this theorem. I am looking for examples which have a real impact on students.

Homework Statement If a , b, c are distinct and p(x) is a polynomial in x which leaves remainders a,b,c on division by (x-a),(x-b),(x-c) respectively. Then the remainder on division of p(x) by(x-a)(x-b)(x-c) is Homework Equations As it is given that p(x) gives remainder a when divided by...

Homework Statement Find all integers x such that 7x \equiv 11 mod 30 and 9x \equiv 17 mod 25 Homework Equations I guess the Chinese Remainder theorem and Bezout's theorem would be used here. The Attempt at a Solution I can do this if the x-terms didn't have a...

Is there a generalization for the Chinese Remainder Theorem if the modular bases are not coprime? Or even to some extent, if the modular bases are increasing by the same common ratio? I searched it up but could not find anything.

If I was to try to work this out I would use the Chinese Remainder Theorem and since 10557 = 3^3 . 17 . 23 end up with (Z/10557Z)* isomorphic to (Z/27Z)* x (Z/17Z)* x (Z/23Z)* isomorphic to C18 x C16 x C22 where Cn represents the Cyclic group order n. How would I then write this as Cn1 x Cn2...

Homework Statement A) Find the remainder of 2^n and 3^n when divided by 5. B)Conclude the remainder of 2792^217 when divided by 5. C)solve in N the following : 1) 7^n+1 Ξ 0(mod5) 2) 2^n+3^n Ξ 0(mod5) The Attempt at a SolutionA) I know that for the first two I have to get 2^n=5k+r and...

1. Prove that the MacLaurin series for cosx converges to cosx for all x. Homework Equations Ʃ(n=0 to infinity) ((-1)^n)(x^2n)/((2n)!) is the MacLaurin series for cosx |Rn(x)|\leqM*(|x|^(n+1))/((n+1)!) if |f^(n+1)(x)|\leqM lim(n->infinity)Rn=0 then a function is equal to its Taylor series...

I have to prove this problem. For all n integers, if n mod 5 = 3, then n2 mod 5 = 4 Proof: Let n be an integer such that n mod 5 = 3. n = 5k+3 for some integer k by definition of MOD or QRT? Which one would be correct? Am I using the definition of MOD or QRT? I'm thinking its QRT because its...

Homework Statement when f(x) is divided by (x+1), remainder is -9; when f(x) is divided by (x-3), remainder is -1; what is the remainder if f(x) is divided by (x+1)(x-3)? Homework Equations f(x) = divider * q(x) + remainder The Attempt at a Solution f(x) = (x+1) * a(x) -9 f(x) =...

So we had two examples in class, but I don't understand why they're different. And the professor is away today, which means I won't see him until the entire weekend has passed (a nightmare for students like me who obsess over a problem). 1. For which x is the approximation sin(x) ≈ x - (x^3)/6...

Homework Statement Prove that for every pair of numbers x and h, \left|sin\left(x+h\right)-\left(sinx+hcosx\right)\right|\leq\frac{h^{2}}{2} The Attempt at a Solution Let f(x)= \left|sin\left(x+h\right)-\left(sinx-hcosx\right)\right|? and then to center the taylor polynomial around 0...

Is there anywhere in topology where one would see the Chinese Remainder Theorem?

Homework Statement The remainder theorem can't really be applied when dividing by something other than a linear equation since you wouldn't know what a is, right? Homework Equations The Attempt at a Solution

The Chinese remainder theorem tells us that the system of equations: \begin{align} x &\equiv a_1 \pmod{n_1} \\ x &\equiv a_2 \pmod{n_2} \\ &\vdots \\ x &\equiv a_k \pmod{n_k} \end{align} Uniquely determines all numbers in the range: X<N=n_1n_2\ldots n_k and that all solutions are...

Homework Statement Find the indicated value of the polynomial using the Remainder Theorem p(x)=2x^3-2x^2+11x-100; find p(3) Homework Equations p(x)=2x^3-2x^2+11x-100 The Attempt at a Solution Synthetic division 3] 2 -2 11 -100 6 12 69 2 4 23 [-31 answer: p(3)=-31 im not...

So I am working on solving sets of linear congruence with the chinese remainder theorem. When I go to solve for the inverses I am meeting a bit of trouble. What do I do when the a term is larger that m? Example 77x=1(mod3) 33y=1(mod7) 21z=1(mod11) where x,y,z are the inverses I am trying...

(a) Let R and S be rings with groups of units R∗ and S ∗ respectively. Prove that (R × S)∗ = R∗ × S ∗ . (b) Prove that the group of units of Zn consists of all cosets of k with k coprime to n. Denote the order of (Zn )∗ by φ(n); this is Euler’s φ-function. (c) Now suppose that m and n are...

Homework Statement (a)Use Definition 10.8.1 to find the Maclaurin series for f(x) = sinh x. Express your answer using Σ notation. (b) Find the interval of convergence for the series found in part (a). (c) Use the Remainder Theorems 10.7.4 and 10.9.2 to show that the series found in part (a)...

Hello, I am looking into proving that the Chinese Remainder Theorem will work for two pairs of congruences IFF a congruent to b modulo(gcd(n,m)) for x congruent to a mod(n) and x congruent to b mod(m). I have gotten one direction, that given a solution to the congruences mod(m*n), then a...

Homework Statement Prove that for any polynomial function f and number a, there exists a polynomial function g and number b such that: f(x) = (x-a)g(x) + b Homework Equations N/A The Attempt at a Solution Proof: Let P(n) be the statement that for some natural number n, f(x) =...

Homework Statement I understand How to do The remainder Theorem and The factor Theorem but I don't understand what they mean or what they are doing. I don't think I will be able to apply them without knowing what they mean. Can someone explain them to me? Homework Equations The...

Homework Statement Find the remainder when (x^80 - 8x^30 + 9x^24 + 5x + 6) is divided by (x+1) Homework Equations The Attempt at a Solution So I'm not really sure where to start. I tried starting by doing long polynomial division, but I get stuck. How do I start this?

Homework Statement Prove that is m, n, and d are integers and d divides (m-n) then m mod d = n mod d. Homework Equations Quotient Remainder Theorem: Given any integer n and positive integer d, there exists unique integers q and r such that n=dq + r and 0\leqr<d and n mod d = r. The...

Chinese remainder theorem, urgent! Homework Statement This is an attempt to make the Chinese Remainder Theorem more concrete. Let m = 206 and n = 125. You may use the fact that 89n - 54m = 1. (a) What does the Chinese Remainder Theorem have to say about pairs of residues modulo 206 and...

Homework Statement I am trying to learn the Chinese Remainder Theorem from the following website: http://www.libraryofmath.com/chinese-remainder-theorem.html The only thing I don't understand is why the end result is expressed as another linear congruence. In the first example, the...

Hi, I can not see how this is implied... Let m and n be positive integers, with gcd(m, n) = 1. The the system of congruences x = a (mod m) and x = b (mod n ) has a solution. Moreover, any two solutions are congruent modulo mn. pf. Since gcd(m,n) = 1, there exist integers r...

This doesn't actually require the use of the CRT, since it actually wants you to sort of derive it for a system of two equations. So while using the CRT will help me solve this fairly quickly and easily, that's not what I'm after Homework Statement Let gcd(m,n)=1. Given integers a,b, show...

I know how the polynomial remainder theorem works but I can't see how knowing this is useful in any way. So I have f(X). I know that if I divide the statement in f(X) by X - a the remainder will be a. How is this useful knowledge though? What can I discover using this principle that I wouldn't...