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. W

    Brainteaser: remainder of 11^345678 / 13

    I stumbled upon this seamingle impossible question (without calculator!), any ideas to find the remainer of \frac{11^{345678}}{13}?
  2. D

    The Chinese Remainder Theorem (the CRT)

    Find the lowest number that has a remainder of 1 when divided by 2, 2 when divided by 3, 3 when divided by 4, 4 when divided by 5, and 5 when divided by 6. It is possible to solve this by applying the general algorithm that solves Chinese Remainder problems. But, for this special...
  3. E

    Finding Remainder of Shaded Squares in 6x4 Grid

    Homework Statement In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let be the number of shadings with this property. Find the remainder when is divided by 1000. There is a...
  4. T

    Polynominal and the remainder theorom

    Homework Statement Using the remainder theorem solve \frac{8x^{3} + 2x^{2} + 5}{2x^{2}+2} Homework Equations The remainder theorem: F(x) \equiv Q(x) \times divisor + remainder The Attempt at a Solution 8x^{3} + 2x^{2} + 5 \equiv (Ax+B)(2x^{2}+2) + C I can do it for...
  5. C

    Where did the six come from in the Chinese Remainder Theorem?

    I need help making sense of my notes: x congruent 4 mod 11 x congruent 3 mod 13 ai mi Mi yi aiMiyi 4 11 13 6 4*13*6 3 13 11 6 3*11*6 I'm not sure where the six came from
  6. B

    Upper Bound Error for Maclaurin Polynomial of Sin(x) on the Interval [0,2]

    Homework Statement Find the 3rd-order Maclaurin Polynomial (i.e. P3,o(u)) for the function f(u) = sin u, together with an upper bound on the magnitude of the associated error (as a function of u), if this is to be used as an approximation to f on the interval [0,2]. I did the question...
  7. N

    Remainder Theorem with 2 unknowns.

    Homework Statement When rx^3 + gx^2 +4x + 1 is divided by x-1, the remainder is 12. When it is divided by x+3, the remainder is -20. Find the values of r and g. Homework Equations The Attempt at a Solution r=f(1) =r(1)^3 + g(1)^2 + 4(1) +5 =r + g +9 r=12 r+g+9=12 r+g= 3...
  8. D

    How Do You Solve Polynomial Equations Using the Factor and Remainder Theorem?

    A couple of hard questions about the Factor and Remainder Thoerem that I'm having a hard time with. Homework Statement 18) f(x) = 2x3 + x2 – 5x + c, where c is a constant. Given that f(1) = 0, (a) find the value of c (b) factorise f(x) completely, (c) find the remainder when...
  9. E

    Proving C(n,m) is an Integer: Number Theory & Chinese Remainder Theorem?

    Homework Statement How would you prove using number theory that C(n,m) is an integer where n => m =>1? Do you need the Chinese Remainder Theorem? It seems like it should follow easily from what C(n,m) represents but it is hard for me for some reason. Homework Equations The Attempt...
  10. T

    Discrete Mathematics with possible Quotient Remainder Theorem

    Homework Statement For all integers m, m^{}2=5k, or m^{}2=5k+1, or m^{}2=5k+4 for some integer k. Relevant equations I'm pretty sure we have to use the Quotient Remainder THM, which is: Given any integer n and positive integer d, there exists unique integers q and r such that...
  11. P

    Find the remainder of a division

    Hi all! Could anyone help telling me the way to find the remainder of the following divisions: 1. (x^2006+x^1996+x^1981+x+1):(x^2-1) 2. (x2+x3+x5+1) : [(x-1)(x-2)] Thanks
  12. mattmns

    Number Theory: Inverse of 0 mod n? Chinese Remainder Theorem

    I am doing a Chinese remainder theorem question and one of the equations is x \equiv 0 (mod 7). This would mean that x is a multiple of 7, but how do I use it in conjunction with the Chinese remainder theorem? Do I just ignore that equation, use the CRT on the rest of the system, and then once...
  13. S

    Find a Degree Three Polynomial with x-2 Remainder of 3 - Hint: Work Backwards!

    Find a polynomial of degree three that when divided by x - 2 has a remainder of 3. You will really have to think on this one. Hint: Work backwards! ok here's the thing I've tried I've looked at other problems but I can barely work problems forward, backwards...well your talking to me here my...
  14. P

    Find the remainder of the equation

    The task is to find the remainder of the equation: \frac{18^2+2^{100}}{11} Now I know that if a \equiv b\ (mod\ m),\ c \equiv d\ (mod\ m) \Rightarrow a + c \equiv b +d\ (mod\ m) and ac \equiv bd\ (mod\ m) so 18^2 \equiv b\ (mod\ 11) \Rightarrow \frac{18^2}{11}=29.454545... \Rightarrow...
  15. Ahmed Abdullah

    Finding the Remainder in Division of Polynomials | Step-by-Step Solution

    What is the remainder?? Homework Statement What is the remainder when (a+b+c)^333-a^333-b^333-c^333 is divided by (a+b+c)^3-a^3-b^3-c^3? Homework Equations None The Attempt at a Solution I tried this (a+b+c)^333-a^333-b^333-c^333 = Q{(a+b+c)^3-a^3-b^3-c^3}+h where h is the...
  16. R

    Remainder Estimation Theorem & Maclaurin Polynomials :[

    Homework Statement Use the Remainder Estimation Theorem to find an interval containing x=0 over which f(x) can be approximated by p(x) to three decimal-place accuracy throughout the interval. Check your answer by graphing |f(x) - p(x)| over the interval you obtained. f(x)= sinx p(x)=...
  17. M

    Why is my BigInteger Program Not Producing Any Output?

    Okay, I made this program in order to solve this question What is the smallest positive integer x such that x^2 + 3x + 5 is divisible by 121? The program complies perfectly. But when I execute it, it don't print out any answer. import java.math.BigInteger; public class Number37 { public...
  18. E

    Solve the Remainder Theorem with x^2-4x^2+3

    remainder theorem...? Find the value of 'a' and 'b' and the remaining factor if the expression ax^3-11x^2+bx+3 is divisible by x^2-4x^2+3 do i simplify x^2-4x^2+3 and then substitute for x? im so lostt!
  19. C

    How many terms do we need to sum for accurate result in alternate series?

    How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}} , | error | < 0.01 . So, b_{n} = \frac{1}{n^{2}} . b_{n} < b_{n+1} , and \lim_{n\rightarrow \infty} b_{n} = 0 . Therefore, the series is...
  20. B

    Division with polynomials involving the remainder theorem

    Two questions, first, I solved something, but I was just playing around with the numbers, and I didn't really know what I was doing, nor did I really understand it after I was done. The question is as follows: When x + 2 is divided into f(x), the remainder is 3. Determine the remainder when x...
  21. C

    Chinese Remainder Theorem: A Powerful Tool in Number Theory

    Chinese Remainder Theorem! I'm pretty sure that the following is in fact the Chinese remainder Theorem: If n= (m1)(m2)...(mk) [basically, product of m's (k of them)] where each m is relatively prime in pairs, then there is an isomorphism from Zn to ( Zm1 X Zm2 X ... X Zmk). Zn...
  22. B

    Finding the Remainder of 111111222222 Divided by 7

    Can anyone help me? I'm trying to find the remainder when 111111222222 is divided by 7 without using any long division. I thinking that i can take the alternating sums of the 3 digits blocks and set that mod7. I'm not sure if I'm on the right track or not. thanks
  23. D

    Find the remainder of a polynomial

    7a) f(x0 = 2x3+32-6x+1 Find the remainder when F(x) is divided by (2x-1) b) (i) Find the remainder when f(x) is divided by (x+2) (ii) Hence, or otherwise solve the equation: 2x3+32-6x-8=0 giving your answer to two decimal places. This is what I've done: a) r= -1 (i did f(1/2) to get...
  24. V

    What is the remainder when 23 raised to 98 is divided by 98?

    Q. What is the remainder when 23 raised to 98 is divided by 98?Why?
  25. B

    Chinese remainder theorem problem

    I'm having a lot of trouble setting up the equations for the following question where I need to use the chinese remainder theorem. Q. Fifteen pirates steal a stack of identical gold coins. When they try to divide them evenly, two coins are left over. A fight erupts and one of the pirates is...
  26. D

    Find the remainder when 1/f(x) is divided by x+1

    It is given that f(x)=8x^3+4x-3. The question is: Find the remainder when 1/f(x) is divided by x+1. My textbook says the remainder does not exist? I just can't solve it. Thanks in advance for any help. Abdullah
  27. M

    What is the remainder when x^X^x^x... is divided by x-700^(1/700)?

    whats the remainder when x^X^x^x... is divided by x-700^(1/700) leaving answer in whole number
  28. C

    Questions on Remainder & Integer Divisibility

    I have 2 questions. 1)what is the remainder with 100! is divided by 103? explain your answer 2)a = 238000 = 2^4 x 5^3 x 7 x 17 and b=299880 = 2^3 x 3^2 x 5 7^2 17. Is there an integer so that a divides b^n? if so what is the smallest possibility for n? the first one i have no...
  29. B

    What happens to the remainder of the energy

    I'm not really sure about such things, but this is my question. :smile: Say a photon of white light in incident on an object which is instrinically a blue object, just any regular blue object. Now (naturally) you see light (a photon corresponding with a wavelength that matches the blue...
  30. M

    Solving a Polynomial Division Problem: Remainder = 5 & 7

    Given that a polymial p(x) is p(x)= (x-1)(x-2) q(x) + 2x+3 where q(x) is also a polynomial Find the remainder when p(x) is divided by (x-1)(x+2) where the remainder divided by (x-1) and (x+2) is both 5 and 7 respectively. I don't know even where to start ! so please help, thanks alot.
  31. J

    Inverse chinese remainder theorem

    hi all I am new on the forum I wonder if is possible to find a method that proofs that a number IS NOT a solution of a set of congruences Maybe using the chinese remainder theorem?? best regards japam
  32. tandoorichicken

    What is the Value of k if the Remainder of p(x) Divided by x-1 is 12?

    If p(x) = (x+2)(x+k) and if the remainder is 12 when p(x) is divided by x-1, then what is the value of k?
  33. H

    How can I prove this using the remainder theorem?

    If p^2 is exactly divisible by p+q, then proof q^2 is exactly divisible by p+q. How do I proof this, and how do I apply the remainder theorem? I know if f(x) = x^2 + 2x + 1, since f(-1) = 0 there fore (x + 1) is a factor of f(x). So in this case p^2 = p x p or p^2 x 1...
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