What is Remainder: Definition and 183 Discussions

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.

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  1. S

    B Remainder when the dividend is less than the divisor

    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...
  2. M

    Find the remainder when ## 823^{823} ## is divided by ## 11 ##

    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...
  3. M

    Finding Integer with Chinese Remainder Theorem

    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...
  4. M

    Find the remainder when ## 4444^{4444} ## is divided by ## 9 ##.

    Observe that ## 4444\equiv 7\pmod {9} ##. This means ## 4444^{4444}\equiv 7^{4444}\pmod {9}\equiv 7^{4+40+400+4000}\pmod {9} ##. Now we have \begin{align*} &7^{4}\equiv 7\pmod {9}\\ &7^{40}\equiv (7^{4})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv [(7^{4})^{2}\cdot 7^{2}]\pmod {9}\equiv...
  5. M

    What is the remainder when the following sum is divided by 4?

    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 ##...
  6. chwala

    Use Remainder theorem to find factors of ##(a-b)^3+(b-c)^3+(c-a)^3##

    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...
  7. Leo Liu

    [IntroNumTheory] Determining the remainder by using congruence

    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...
  8. S

    Bounds of the remainder of a Taylor series

    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|...
  9. M

    MHB Value of f(1/2) using estimation for the remainder

    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) ...
  10. S

    Remainder of polynomial division

    ##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
  11. S

    Remainder of polynomial

    ##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
  12. chwala

    Factor and remainder theorem problem

    ##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##
  13. TonkaTon

    A gas stream (1) contains 18 mol% hexane and remainder nitrogen flows

    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...
  14. Adgorn

    Limit of the remainder of Taylor polynomial of composite functions

    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...
  15. S

    Remainder of a polynomial

    f(x) = A(x) . (x2 + 4) + 2x + 1 f(x) = B(x). (x2 + 6) + 6x - 1 f(x) = C(x) . (x2 + 6) . (x2 + 4) + s(x) Then I am stuck. What will be the next step? Thanks
  16. V

    MHB How to solve Chinese Remainder Theorem

    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...
  17. Monoxdifly

    MHB Find Polynomial Given Remainder After Division

    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...
  18. Poetria

    Approximating square root of 2 (Taylor remainder)

    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...
  19. F

    Chinese remainder theorem (CRT) question, flat shapes

    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...
  20. F

    Chinese Remainder theorem for 2 congruences

    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...
  21. S

    MHB About a variant of the Chinese Remainder Theorem

    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...
  22. D

    MHB Find Remainder When Divided by 19

    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
  23. M

    MHB Remainder converges uniformly to 0

    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 $0$...
  24. M

    MHB Taylor polynomial - remainder

    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...
  25. S

    B Remainder of polynomial division

    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
  26. J

    Remainder Theorem problem

    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!
  27. A

    MHB An inequality between the integral Remainder of a function and the function.

    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...
  28. B

    Ring in which Quotient and Remainder not Unique

    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...
  29. V

    Can anyone me with this division and remainder problem?

    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)
  30. Vital

    Finding the value based on the value of the remainder

    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...
  31. Y

    Calculating Remainders: Solution to (1*1!+2*2!+...+12*12!) / 13

    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
  32. kaliprasad

    MHB Find remainder when (2∗4∗6∗8⋯∗2016)−(1∗3∗5∗7⋯∗2015) is divided by 2017

    $ ( 2 * 4 * 6 * 8 \cdots * 2016) - ( 1 * 3 * 5 * 7 \cdots * 2015)$ is divided by 2017 what is the remainder
  33. T

    Remainder factor theorem: me reason this out

    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...
  34. Schaus

    Using Remainder Theorem to find remainder

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

    I Positive or negative remainder

    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?
  36. T

    Using remainder factor theorem

    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...
  37. E

    Application of Fermat's Little Theorem

    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...
  38. Albert1

    MHB Find Remainder of $40^{110}$ and $3^{1000}$ Divided by 37 and 26

    (1)dividing $40^{110} \,\, by \,\, 37$ (2)dividing $3^{1000} \,\, by \,\, 26$
  39. FritoTaco

    Long Division and Remainder Theorem

    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 -...
  40. S

    A tricky remainder theorem problem

    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) +...
  41. terryds

    Remainder theorem Polynomial

    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]...
  42. S

    B Is there a lemma named for this?

    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...
  43. I

    I Alternative Ways To Find The Remainder of A/B?

    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 } $$\frac{A}{B} , A > B$$
  44. NoName3

    MHB Understanding the Chinese Remainder Theorem for $\mathbb{Z}^{\times} _{20}$

    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...
  45. G

    MHB Hard remainder problem

    Find ${5^{2009}}^{1492}\mod{503}.$ How do you calculate a beast like this?
  46. G

    MHB Derive Taylor Expansion of $f(x)$ Around x=0 Up to x^2

    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?
  47. T

    MHB Quotient remainder theorem

    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?
  48. T

    MHB Quotient remainder theorem problem.

    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.
  49. P

    MHB Largest remainder method

    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...
  50. Julio1

    MHB Problem Chinese remainder Theorem

    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?
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