Remainder theorem Definition and 71 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 By hand, find the 4 square roots of 340 mod 437. (437 = 23 * 19). Homework Equations Chinese remainder theorem (CRT) The Attempt at a Solution So this is the wrong way I did it was first I solved ##x^2 \equiv 340 (\operatorname{mod} 19)## and ##x^2 \equiv 340...

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?

Show that if $\text{gcd}(b,c)=1$, then $\forall r,s\in \mathbb{N}, \exists x\in \{1,...,bc\}$ such that $x\in (r+b\mathbb{N})\cap (s+c\mathbb{N}).$ Hello :). Can define an function $\varphi: \{1,...,bc\}\to \mathbb{Z}/b\mathbb{Z}\times \mathbb{Z}/c\mathbb{Z}$ at follow $x\mapsto ([x]_b,[x]_c)$...

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 Solve x \cong 1 mod 7 x \cong 4 mod 6 x \cong 3 mod 5 by (and I have to use this method) using Euclidean algorithm to find the largest common divisor, then the extended euclidean algorithm to find a linear combination, then a solution to each of the three...

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) =...

Say I have a^27,654,321 modulo 100,000,000 (where Euler's Theorem no longer helps because totient(100,000,000) = 40,000,000 which is larger than my exponent). How do I use the Chinese Remainder Theorem here to shrink down my massive a^b term I have here? It seems like I need to first split up...

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...

Given system of congruences, x^3 \equiv y_1 \mod n_1 x^3 \equiv y_2 \mod n_2 x^3 \equiv y_3 \mod n_3 You are given y_1, y_2, y_3, n_1, n_2, n_3. The n_i's are pairwise relatively prime. Solve for x. I think there might be a connection between the fact that the exponent of x is 3 and...

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

Homework Statement Find a satisfying a 1 (mod 2), a 2 (mod 3), with 0 < a < 6 Begin with the Mod 2 calculation Homework Equations The Attempt at a Solution I found that U1=3 and U2=4, but then it says "Now take the appropriate linear combination of {U1, U2} to...

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 I am looking for some help in finding the Lagrange Remainder Theorem from the integral form of the remainder of a Taylor series Homework Equations Integral form of Taylor Series: Rn,a(x) = x∫a [f(n+1)(t)]/n! *(x-t)dt The Attempt at a Solution We are given the...

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...