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. J

    Eigenvalues and eigenvectors, 3x3 matrix using Remainder and Factor Theorem

    Homework Statement i = the 3x3 matrix below 2-λ 0 1 -1 4-λ -1 -1 2 0-λ Using remainder and factor theorem find the 3 values of λ. Homework Equations |i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3| |a|=ad-bc The Attempt at a Solution(2-λ) |(4-λ x...
  2. B

    Remainder estimate for product series

    Suppose I have two series A=\sum_{n=0}^\infty a_n B=\sum_{n=0}^\infty b_n and I have estimates for the remainders of each one: \sum_{n=N}^\infty a_n \le R^N_A \sum_{n=N}^\infty b_n \le R^N_B Consider the product series AB=\sum_{n=0}^\infty c_n where c_n=\sum_{i=0}^n a_i...
  3. P

    Question about the remainder of a division

    I was self-studying Discrete Mathematics and I found the following problem. Homework Statement Show that whenever a and b are positive integers, then (2^a-1)\mod(2^b-1)=2^{a\mod b}-1 The attempt at a solution I don't know if there is a more rigorous way of solving it, but I came up with the...
  4. S

    Analysis problem using the Lagrange Remainder Theorem

    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...
  5. Saitama

    What is the remainder when 4101 is divided by 101?

    Homework Statement Hi! This is not a homework question but it came in my test. What is the remainder when 4101 is divided by 101? Homework Equations (1+x)n-1 is always divisible by x. The Attempt at a Solution Only the equation given above came in my mind but i wasn't able to...
  6. B

    Division of integer square by 4 leaves remainder 0 or 1

    Hi, I am looking for an explanation, if any, on why every integer square leaves remainder 0 or 1 on division by 4. Appreciate your time and help bluemoon2188
  7. S

    Calculating Remainders for Taylor Series of Sine Function

    Usually to do the remainder we take Rn(x) = (f differentiated n+1 times at a ).(x-c)n+1/(n+1)!, but when my function is sin(x) do i take (f differentiated 2n+2 times at a ).(x-c)2n+2/(2n+2)!? Thanks
  8. D

    Would much appreciate help to figure the remainder of 10^100mod 1001

    Homework Statement struggling to understand how to solve this question and would be extrememly grateful if someone who understood offered me a hand: need to find the remainder of gogool divided by 1001. i.e 10^100mod1001. any assistance would be much welcomed as i have no hope figuring...
  9. E

    What is the value of k for which (a-3b) is a factor of a4 - 7a2b2 + kb4?

    Find the value of k for which (a-3b) is a factor of a4 - 7a2b2 + kb4. Hence, for this value of k, factorize a4 - 7a2b2 + kb4 completely. I tried to do it but my mind is not going anywhere. Any help will be greatly appreciated. :)
  10. P

    Chinese Remainder Theorem (I think?)

    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...
  11. M

    Number Theory: Division with remainder of factorials

    I'm struggling with how to even begin with this problem. Find the remainder of the division of 75!*130! by 211. 211 is prime, so I know the remainder is not 0. I'm not sure where to start though. Thanks!
  12. M

    Number Theory - Find Remainder when dividing by 17

    Number Theory -- Find Remainder .. when dividing by 17 Homework Statement Find the remainder when 3^24*5^13 is divided by 17. Homework Equations I know that 3^24 = 16 (mod 17) and calculated that 5^13 mod 17 = 3 (mod 17) The Attempt at a Solution BUT, I'm completely unsure...
  13. R

    Taylor polynomial of f(x) = 1/(1-x) and the estimate of its remainder

    Homework Statement Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder. The Attempt at a Solution I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2]...
  14. F

    Topology and the Chinese Remainder Theorem?

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

    Using power series remainder term

    Homework Statement (For power series about x=1) Using the error formula, show that \left|ln(1.5)-p_{3}(1.5)\right|\leq\frac{(0.5)^{4}}{4} Homework Equations p_{3}(x) = x-1 - \frac{(x-1)^{2}}{2} + \frac{(x-1)^{3}}{3} \\\epsilon_{n}(x)=\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_{o})^{n+1}\\where \xi lies...
  16. Z

    Remainder theorem only works with quadratics divided by linear?

    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
  17. R

    Taylor Polynomials- Lagrange remainder

    So I'm studying for a final, and it just so happens my professor threw taylor polynomials at us in the last week.. I understand the concept of a taylor polynomial but i need some help fully understand the LaGrange remainder theorem if we have a function that has n derivatives on the interval...
  18. M

    How to Apply the Chinese Remainder Theorem for Congruences?

    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...
  19. jegues

    Small confusion about Taylor's remainder

    Homework Statement Please assume that what I have for the remainder is correct, and we are on the domain 2 < x < 4 around 2. Homework Equations The Attempt at a Solution 0 \leq |R_{n} (2,x)| = \frac{1}{n+1} |\frac{x-2}{z_{n}}|^{n+1} Since, 2 < x < 4 then, 0 < x-2 < 2...
  20. jegues

    Explaining Taylor's Remainder Formula for Convergence of Taylor Series

    Homework Statement Use Taylor's remainder formula to show that the Taylor series for f(x) is about the point indicated converges to f(x) for all x. f(x) = e^{5x} about x=0 Homework Equations The Attempt at a Solution Since, f^{n}(x) = 5^{n}e^{5x}, Taylor's remainder...
  21. J

    Find and Test Primes using the Chinese Remainder Theorem and Binary Search

    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...
  22. T

    Balance of Integer Value: Solving (2^1000) / 7 for Remainder | Homework

    Homework Statement How much is the balance if u want to get integer value of ((2^1000) divided by 7)) Homework Equations The Attempt at a Solution I need a little hint to start off with an attempt.
  23. F

    Finding value of polynomial using the remainder theorem

    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...
  24. S

    Finding the Maximum Remainder in a Taylor Series: Explained

    Hello, I was wondering if anyone could explain to me the thought process behind how you find the maximum remainder of a Taylor series? I read the wiki article and didn't help me at all, http://en.wikipedia.org/wiki/Taylor's_theorem My book talks about something like this(image is...
  25. C

    How Many Terms Needed for Accurate Series Estimation with Less Than 0.001 Error?

    Homework Statement Okay, well this was a question on one of my recent tests: How many terms do you have to use to estimate the sum from n = 0 to n = infinity of (-e/pi)^n with an error of less than .001? Homework Equations Alternating series remainder theorem: For an...
  26. C

    Chinese Remainder Theorem, Solving For Multiplicative Inverses

    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...
  27. R

    Chinese remainder theorem

    (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...
  28. D

    Proving 10^n Leaves Remainder 1 When Divided by 9

    Homework Statement Prove that 10^n leaves remainder 1 after dividing by 9. The Attempt at a Solution There is an integer K, such that 10^n = 9k + 1 Where do i go from here if I want to do it just directly?
  29. I

    Understanding the Remainder Term in Taylor Series: A Closer Look at the Formula

    I'm currently studying the Taylor series and I cannot figure out how the remainder term came to be. If anyone could clarify this for me, I would be really grateful ...! I understand that the Taylor series isn't always equal to f(x) for each x, so we put Rn at the end as the remainder term...
  30. F

    Mac Series f(x)=sinh (x), remainder theorem help

    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)...
  31. A

    The Chinese Remainder Theorem for moduli that aren't relatively prime

    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...
  32. H

    Optimal Degree for Approximating Cosine with Taylor Series

    Homework Statement What degree Taylor Polynomial around a = 0(MacLaurin) is needed to approximate cos(0.25) to 5 decimals of accuracy? Homework Equations taylor series...to complicated to type out here remainder of nth degree taylor polynomial = |R(x)| <= M/(n+1)! * |x - a|^(n+1)...
  33. S

    Finding the Lagrange Remainder Theorem from the Integral Form of a Taylor Series

    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...
  34. A

    Polynomial, Division, Remainder

    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+2)(x-1). EDIT: Took out my attempts lol, there were way off. This was a "Math...
  35. A

    Proof of the Remainder Theorem: Is this proof of the Remainder Theorem valid?

    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) =...
  36. S

    The Remainder Theorem and The Factor Theorem

    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...
  37. W

    Remainder of the product of the relatively prime numbers

    Hi all, I had a problem, pls help me. Let b_1 < b_2 < \cdots < b_{\varphi(m)} be the integers between 1 and m that are relatively prime to m (including 1), of course, \varphi(m) is the number of integers between 1 and m that are relatively prime to m, and let B =...
  38. J

    Estimate Remainder of Taylor Series

    1. The problem \statement, all variables and given/known data Estimate the error involved in using the first n terms for the function F(x) = \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I am using the Lagrange form of the remainder. I need to know the n+1 derivative of...
  39. D

    Remainder Theorem: Solve x^80 - 8x^30 + 9x^24 + 5x + 6 Divided by (x+1)

    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?
  40. D

    Remainder of 34! divided by 37 using Wilson's Theorem

    Homework Statement Find the remainder when 34! is divided by 37. Homework Equations Wilson's Theorem The Attempt at a Solution I understand that (p-1)! = (-1)(mod p) and that (p-2)! = (1)(mod p). I don't understand how to apply this to (p-3)! though.
  41. C

    LaGrange Remainder Infinite Series

    Homework Statement Let f be a function whose seventh derivative is f7(x) = 10,000cos x. If x = 1 is in the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at x = 0 will approximate f(1) with an error of not more than a.)...
  42. N

    Maximum Error in Taylor Polynomial for cos(x) on Interval [-.25, .25]

    Homework Statement Find the maximum error in approximating cos(x) by its Taylor polynomial of order 2 on the interval [ −.25, .25]. Justify your answer using the Remainder Estimation Theorem. Homework Equations |R3(x)<=M/3! |x|^3 The Attempt at a Solution |R3(x)<=M/3! |x|^3...
  43. C

    Proving m mod d = n mod d with Quotient Remainder Theorem

    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...
  44. S

    Chinese Remainder Theorem: How Can It Help Solve Modulo Residue Problems?

    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...
  45. S

    What is the solution expressed in the Chinese Remainder Theorem?

    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...
  46. D

    Yes, that was exactly what I couldn't get. Thank you for clarifying it for me!

    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...
  47. B

    Chinese Remainder Theorem

    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...
  48. K

    Using the Lagrange Remainder Theorem (advanced calc/real analysis)

    Hi everyone. This is my first post here and I was wondering if any of you could help me. The question is to prove that 1 + \frac{x}{3} - \frac{x^2}{9} < (1 + x)^\frac{1}{3} < 1 + \frac{x}{3} if x>0. The question is in a section on the lagrange remainder theorem. The fact that the first...
  49. M

    How can the Polynomial Remainder Theorem be applied in real-life situations?

    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...
  50. V

    Am I right or is my book right (Taylor remainder)

    Homework Statement The approximation e^{x}=1+x+(x^{2}/2) is used when X is small estimate the error when \left|x \right|<0.1Homework Equations \left|R_{n} \right|<\frac{M(x-a)^{n+1}}{(n+1)!}The Attempt at a Solution Since the Taylor expansion goes to the second power I used the third...
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