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. Julio1

    MHB Can $\varphi$ be used to prove the Chinese remainder theorem?

    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 $x\mapsto ([x]_b,[x]_c)$...
  2. M

    MHB What is the Remainder When \(121^{103}\) is Divided by 101?

    What is the remainder of the division number $121^{103}$ by 101
  3. Pull and Twist

    MHB Remainder Estimate for Integral Test

    I'm working on the following problem and I have made it this far... am I on the correct path or am I doing this incorrectly?? I find series extremely confusing. Also... how do I find the error involved in the improved approximation? This is the series I am working with...
  4. B

    Estimates of the remainder in Taylor's theorem

    Here is the exercise question; Use the general binomial series to get ##\sqrt{1.2}## up to 2 decimal points In the solution the ##R_1## was given as ##|R_1|\leq {\frac{1}{8}} {\frac{(0.2)^2}{2}}## But it doesn't say where this came from and comparing this with the estimate of remainder given in...
  5. M

    How can the Lagrange remainder theorem be applied to series with skipped terms?

    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...
  6. A

    Calculating Taylor Series Remainder: Finding an Upper Bound for n

    How is the Taylor remainder of a series (with given Taylor expansion) expressed if you want to make a calculation with known error? e.g. if I want to calculate π to, say, 12 decimal places using the previously-derived result π=4*arctan(1) and the Taylor series for arctan(x), how will I work out...
  7. snoopies622

    Finding the remainder of an algebraic quotient

    I'm tutoring a pupil for a CLEP exam and her book includes the following algebra problem: What is the remainder when 9x^{23} - 7x^{12} - 2x^{5} +1 is divided by x+1 ? I know how to find the answer by computing the quotient of these two expressions, but in this case doing that is so tedious I...
  8. RJLiberator

    Evaluating the remainder of a Taylor Series Polynomial

    Homework Statement The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1. Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx...
  9. 22990atinesh

    How to find Remainder of this expression?

    Homework Statement Q. How to find Remainder of the following expression ##32^{32^{32}}\mod{}6=?## Homework Equations The Attempt at a Solution ##32^{32}\mod{}2=0 \implies 32^{32}=2x## ##32^{32}\mod{}3=1 \implies 32^{32}=3y+1## would it help to find the remainder of ##32^{32^{32}}\mod{}6##
  10. R

    Modulus & Division: Last Digit of Numbers Explained

    Isn't it amusing ?What could be the probable explanation for this?Also when operated by division operator gives the rest of the number as the quotient (Note only when the divisor is 10)
  11. Albert1

    MHB Remainder of $\dfrac {19^{81}+19^{49}+19^{25}+19^9+19}{19^3-19}$

    please find the remainder $\dfrac {19^{81}+19^{49}+19^{25}+19^9+19}{19^3-19}$
  12. PsychonautQQ

    Proof of Chinese Remainder Theory

    Homework Statement Theorm: Let m and n be relatively prime integers. If s and t are arbitrary integers there exists a solution x in Z to the simultaneous congruences: x~s (mod m) and x~t (mod n) Part of proof that confuses me: Since gcd(m,n) = 1, the Euclidean algorithm gives p and q in...
  13. kaliprasad

    MHB Remainder of $3^{2^n}-1$ Divided by $2^{n+3}$

    find the remainder when $3^{2^n}-1$ is divided by $2^{n+3}$
  14. B

    Points of Convergence for Lagrange Remainder Theorem

    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}...
  15. Saitama

    MHB What is the remainder when $3^{2002}+7^{2002}+2002$ is divided by 29?

    Problem: When $3^{2002}+7^{2002}+2002$ is divided by 29, the remainder is: A)0 B)1 C)2 D)7 Attempt: I tried the following: $3^3 \equiv -2\mod 29$ i.e $3^{2002} \equiv 3\cdot 3^{2001} \equiv -3\cdot 2^{667} \mod 29$ Also, $2^5 \equiv 3 \mod 29$, hence, $-3\cdot 2^{667} \equiv -3\cdot...
  16. S

    Difficult polynomial question involving factor and remainder theorems

    Homework Statement Prove that ##(a-b)## is a factor of ##a^5-b^5##, and find the other factor. Homework Equations Remainder theorem : remainder polynomial ##p(x)## divided by ##(x-a)## is equal to ##p(a)## Factor theorem : if remainder = 0, then divisor was a factor of dividend...
  17. J

    MHB How Does the Remainder Theorem Simplify Polynomial Division?

    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.
  18. J

    MHB Factor and remainder theorem question

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

    Alternating Series estimation theorem vs taylor remainder

    Homework Statement Let Tn(x) be the degree n polynomial of the function sin x at a=0. Suppose you approx f(x) by Tn(x) if abs(x)<=1, how many terms are need (what is n) to obtain an error less than 1/120 Homework Equations Rn(x)=M(x-a)^(n+1)/(n+1)! sin(x)=sum from 0 to ∞ of...
  20. A

    Solving Polynomial Remainders

    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.
  21. F

    Remainder Theorem Thinking Question

    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...
  22. matqkks

    What are some practical applications of the 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.
  23. matqkks

    MHB How Can the Chinese Remainder Theorem Be Applied to Diophantine Equations?

    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.
  24. S

    MHB What is the remainder when 1992 is divided by 92 using the CRT?

    Find the remainder when 1992 is divided by 92
  25. R

    Remainder After Division Problem

    Homework Statement (-8)^4124 + 6^3101 + 7^5 is divided by 3. Homework Equations The Attempt at a Solution My original insight was that 6 raised to any power is always divisible by 3. 7 raised to any power yields a remainder of 1 when divided by 3. and the remainder from -8...
  26. M

    Extended euclidean algorithm and Chinese Remainder theorem

    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...
  27. anemone

    MHB What is the remainder when a_{2013} is divided by 7?

    Consider a sequence given by a_n=a_{n-1}+3a_{n-2}+a_{n-3}, where a_0=a_1=a_2=1. What is the remainder of a_{2013} divided by 7?
  28. anemone

    MHB Find the remainder when f(100) is divided by by 100

    Consider the triangular array of numbers 0,\;1,\;2,\;3,\cdots along the sides and interior numbers obtained by adding the two adjacent numbers in the previoius row. Row 1 through row 6 are shown as below. 0 1 1 2 2 2 3 4 4 3 4 7 8 7 4 5 11...
  29. icystrike

    Lagrange Remainder: Clarifying MVT Statement

    Homework Statement This is not a homework problem but I would like to clarify my concern. It is stated that a function can be written as such: f(x) = \lim_{n \rightarrow ∞} \sum^{∞}_{k=0} f^{(k)} \frac{(x-x_{0})^k}{k!} R_{n}=\int^{x}_{x_{0}} f^{(n+1)} (t) \frac{(x-t)^n}{n!} dt They...
  30. S

    What is the Remainder When a Polynomial is Divided by a Product of Linear Terms?

    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...
  31. S

    Applying Chinese Remainder Theorem to polynomials

    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...
  32. E

    Generalization of Chinese Remainder Theorem

    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.
  33. S

    Taylor polynomial remainder term

    Homework Statement Consider the followign function f(x) = x^-5 a=1 n=2 0.8 \leq x \leq 1.2 a) Approximate f with a tayloy polynomial of nth degree at the number a = 1 b) use taylor's inequality to estimate the accuracy of approximation f(x) ≈ T_{n}(x) when x lies in the interval...
  34. P

    Lagrange Remainder for Taylor Expansion of ln(4/5) ≤ 1/1000?

    Hi, Homework Statement I am trying to limit Lagrange's remainder on taylor expansion of ln(4/5) to be ≤ 1/1000. Homework Equations The Attempt at a Solution I have tried using both ln(1+x), where x=-1/5 and x0(the center)=0, and ln(x), where x=4/5 and x0=1. Every time I keep...
  35. C

    Maclaurin remainder interval estimate

    Homework Statement The question asks to estimate the remainder on the interval |x|≤ 1. f(x) is given as sinh(x). I solved the polynomial P3(x) = x + (1/6)(x3) I then went ahead and solved R3(x) up to the point shown below. R3(x) = (sinh(c)*x4)(1/24)I then don't know how to go about...
  36. S

    Polynomial Remainder Therem to proove this

    Homework Statement Applying remainder theorem again and again to show that the remainder of the f(x) polynomial function when divided by (x-α)(x-β) is A(x-α)+B . Determine A and B Homework Equations the remainder of a polynomial f(x), divided by a linear divisor x-a, is equal to f(a) The...
  37. Saitama

    Find Remainder of 25! Divided by 29: Help Appreciated

    Homework Statement Find the remainder of \frac{25!}{29} Homework Equations The Attempt at a Solution One of my friend asked me this question and i was clueless how should i start? (I am not sure that the question is correct.) Any help is appreciated!
  38. R

    (Z/10557Z)* as Abelian Groups using Chinese Remainder Theorem

    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...
  39. M

    Number theory: ( remainder theorem.)

    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...
  40. V

    Proving Lagrange's Remainder from Taylor's Theorem

    Homework Statement I'm interested in the use of the mean value theorem of integration to convert Taylor's theorem into Lagrange's remainder. Though, I'm confused as to how it's used in the conversion. Could someone explain to me how the MVT of integration is used to convert Taylor's theorem...
  41. S

    Taylor Series Remainder Theorem

    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...
  42. I

    Mod or quotient remainder theorem (QRT)

    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...
  43. S

    Solving Series w/ Remainder Estimate & Integral Test

    It asks "use remainder estimate for integral test" to find series accurate to 3 dec? Homework Statement It says "Use the Remainder Estimate for the Integral Test to find the sum of the following series to three decimal places of accuracy." \sum^{\infty}_{n=1} \frac{1}{n^{3}} Homework...
  44. T

    Remainder theorem question - combine divisor

    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) =...
  45. S

    Chinese Remainder Theorem on large exponents

    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...
  46. W

    Proof: Quotient and Remainder involving floor function

    Homework Statement Show that if a (is in) Z and d (is in) Z+, d>1 then the quotient and remainder when a is divided by d are a/d and a-d(floor function(a/d)) Homework Equations The Attempt at a Solution solution (that i have from handout - that i don't understand) by thm 2 p202 (? i am not...
  47. J

    Taylor Polynomial with Remainder Question

    Homework Statement What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places? Homework Equations R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial) The Attempt at a Solution I have attached my attempt. I am stuck on the...
  48. L

    Oblique Asymptotes: What happens to the Remainder?

    Let's say I'm trying to find the oblique asymptote of the function: f(x)= -3x2 + 2 x-1 Forgive my poor formatting. So because the denominator isn't linear, we do polynomial long division of the function and ultimately get -3x - 3 as our quotient, with a remainder of...
  49. C

    Remainder Theorem and Error Help Why are these 2 examples different?

    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...
  50. S

    Remainder for Maclaurin Series

    Homework Statement Find the Maclaurin series of f(x) = x^2cos(x) Homework Equations I got the answer to be (sum from n=1 to infinity) \frac{(-1)(^n+1)x(^2n)}{(2n-2)!} and the formula for the remainder is R_n(x) = \frac{f(^n+1)(c)}{(n+1)!}x(^n+1) (I have no idea how to make those exponents...
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