In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation is the operation that produces such a remainder when given a dividend and divisor.
Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the difference. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion, where the error expression ("the rest") is referred to as the remainder term.
I am Sorry it's a silly question but its been bothering me, So I needed to ask...
In the process of division, when there is a condition like- 4/5
when the divisor is greater then the dividend.
why is the Remainder 4 ? as when the divisor is greater then the dividend then the division goes in...
Observe that ## 823\equiv 9\pmod {11}\equiv -2\pmod {11} ##.
This implies ## 823^{823}\equiv (-2)^{823}\pmod {11} ##.
Applying the Euler-Fermat theorem produces:
## gcd(-2, 11)=1 ## and ## (-2)^{\phi(11)}\equiv 1\pmod {11} ##.
Since ## \phi(p)=p-1 ## where ## p ## is any prime, it follows that...
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...
Let ## n ## be an integer.
Now we consider two cases.
Case #1: Suppose ## n ## is even.
Then ## n=2k ## for some ## k\in\mathbb{N} ##.
Thus ## n^{5}=(2k)^{5}=32k^{5}\equiv 0 \pmod 4 ##.
Case #2: Suppose ## n ## is odd.
Then ## n=4k+1 ## or ## n=4k+3 ## for some ## k\in\mathbb{N} ##.
Thus ##...
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...
https://www.physicsforums.com/attachments/292386
I need to use the congruence to solve this question. My strategy is to write the question as a congruence and then simplify the congruence so that I can apply Congruence to remainder to get the remainder. My work is as follows:
We know that...
I have found the Taylor series up to 4th derivative:
$$f(x)=\frac{1}{2}-\frac{1}{4}(x-1)+\frac{1}{8}(x-1)^2-\frac{1}{16}(x-1)^3+\frac{1}{32}(x-1)^4$$
Using Taylor Inequality:
##a=1, d=2## and ##f^{4} (x)=\frac{24}{(1+x)^5}##
I need to find M that satisfies ##|f^4 (x)| \leq M##
From ##|x-1|...
Hey! :giggle:
Let $f(x)=e^{-x}\sin (x)$, $x\in \mathbb{R}$.
a) Calculate the Taylor polynomial of order $4$ at $0$.
b) Calculate the value of $f \left (\frac{1}{2}\right )$ using estimation for the remainder with an error not more than $\frac{1}{400}$.I have done question a) ...
##x^{2017} + 1 = Q(x) . (x-1)^2 + ax + b## where ##Q(x)## is the quotient and ##ax+b## is the remainder
##x=1 \rightarrow 2 =a+b##
Then how to proceed?
Thanks
##f(x)## is divisible by ##(x-1) \rightarrow f(1) = 0## ##f(x) = Q(x).(x-1)(x+1) + R(x)## where ##Q(x)## is the quotient and ##R(x)## is the remainderSeeing all the options have ##f(-1)##, I tried to find ##f(-1)##:
##f(-1) = R(-1)##
I do not know how to continue
Thanks
Summary: Hello, I need some help with this problem since my professor is bad at explaining (he reads a book and repeats everything), there's a problem online similar, but values and what is asked is different.
A gas stream (1) contains 18 mol% (40.2 mass%) hexane and remainder nitrogen flows...
Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
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...
11. Given a polynomial with the degree 3. If it is divided by x^2+2x-3, the remainder is 2x + 1. If it is divided by x^2+2x, the remainder is 3x - 2. The polynomial is ...
A. \frac23x^3+\frac43x^2+3x-2
B. \frac23x^3+\frac43x^2+3x+2
C. \frac23x^3+\frac43x^2-3x+2
D. x^3+2x^2+3x-2
E. 2x^3+4x^2+3x+2...
Homework Statement
[/B]
Use the Taylor remainder theorem to give an expression of
##\sqrt 2 - P_3(1)##
P_3(x) - the degree 3 Taylor polynomial ##\sqrt {1+x}## in terms of c, where c is some number between 0 and 1
Find the maximum over the interval [0, 1] of the absolute value of the...
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...
Compute the remainder of 2^(2^17) + 1 when divided by 19. The book says to first compute 2^17 mod 18 but I don’t understand why we go to mod 18. Advice would be appreciated
Hey! :o
We have the function $f (x) = e^{\lambda x}$ on an interval $[a, b] , \ \lambda \in \mathbb{R}$.
I want to show that the remainder $R_n (x) = f (x)- p_n (x)$ at the lagrange interpolation of $f (x)$ with $n+1$ points from $[a, b]$ for $n \rightarrow \infty$ converges uniformly to $0$...
Hey! :o
I want to calculate the Taylor polynomial of order $n$ for the funktion $ f(x) = \frac{1}{ 1−x}$ for $x_0=0$ and $0 < x < 1$ and the remainder $R_n$.
We have that \begin{equation*}f^{(k)}(x)=\frac{k!}{(1-x)^{k+1}}\end{equation*}
I have calculated that...
Is this true?
If the remainder of f(x) / g(x) is a (where a is constant), then the remainder of (f(x))n / g(x) is an
I don't know how to be sure whether it is correct or wrong. I just did several examples and it works.
Thanks
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!
Suppose we have a function $f(x)$ which is infinitely differentiable in the neighborhood of $x_0$, and that: $f^{(k)}(x) \ge 0$ for each $k=0,1,2,3,\ldots$ for all $x$ in this neighborhood.
Let $R_n(x)=\frac{1}{n!}\int_a^x f^{(n+1)}(t)(x-t)^n dt$ where $x_0-\epsilon <a<x<b<x_0+\epsilon$;
I...
Homework Statement
Give an example of a commutative ring ##R## and ##f(x), g(x) R[x]## with ##f## monic such that the remainder after dividing ##g## by ##f## is not unique; that is, there are ##q,q',r,r' \in R[x]## with ##qf + r = g = q' f + r'## and ##\deg (r)## and ##\deg (r')## are both...
Is there any chance someone can help me solve this? Music teacher with absolutely no idea how to solve this. Thanl you so much.
? w!81 = 26r3 (the goal is to find the first number, and explain how you figured it out)
Homework Statement
Hello!
Please, help me to learn how to solve the following task - I really have no idea how to do that. What's important is that I need an algorithm that I can apply to the equation with different values.
Homework Equations
The initial equation:
(y - z + i) mod m = x - z...
Homework Statement
What is the remainder when (1*1!+2*2!+...+12*12!) Is divided by 13? Please give the answer along with the steps.
Homework EquationsThe Attempt at a Solution
Homework Statement
find the number of polynomials f(x) that satisfies the condition:
f(x) is monic polynomial, has degree 1000, has integer coefficients, and it can divide f(2x^3 + x)
i would very much prefer that you guys give me hints first. thanks
Homework Equations
remainder factor theorem...
Is 23 = 5(-4)-3 gives a remainder -3 when divided by 5 ? is this statement true ? some of my colleagues said that remainder cannot be negative numbers as definition but I am doubt that can -3 be a remainder too?
1. Homework Statement
i attached the problem statement as an image file
Homework Equations
p(x) = (x-c)q(x) + r
The Attempt at a Solution
i've simplified it down to ((x-1)^114) / (2^114)(x+1). is there a practical way to approach this besides long division? wolfram alpha gave an extremely...
Homework Statement
Find the remainder of ##4^{87}## in the division by ##17##.
Homework Equations
Fermat's Little Theorem:
If ##p## is prime and ##a## is an integer not divisible by ##p##, then
##a^{p-1} \equiv 1 (\mod \space p)##
or equivalently,
##a^p \equiv a (\mod \space p)##
The...
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]...
I'll call it the "Wheel Lug Lemma" for now.
If there are a pair of integers p & q such that the Greatest Common Denominator is 1, and there is some number s that is product of p and an increasing whole number n, then the remainder of the division of s by q will cycle through all values of from...
I guess this would be a Number Theory question. Short of actually going through the division process, is there another way to find the decimal remainder of an arbitrary set of integers { A , B }
$$\frac{A}{B} , A > B$$
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...
Consider the function $f(x) =\sqrt{1 + \sin(x)}$. Derive the fi rst few terms in the Taylor expansion of $f(x)$ around $x = 0$, up to and including terms of order $x^2$. Give an explicit formula for the remainder term.
I can do the first part, but how do I find the remainder term?
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.
Hello All,
I'm working on a problem which uses the largest remainder method
https://en.wikipedia.org/wiki/Largest_remainder_method
I need to allocate a trade quantity among 2 or more strategies.
e.g.
Trade Qty = 99
Strategy A ratio = 0.61
Strategy B ratio = 0.09
Strategy C ratio = 0.23...
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?