The quotient remainder theorem (article) | Khan Academy (2024)
Here's the proof.
Proof of the Quotient Remainder Theorem
We want to prove: 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
We have to prove two things: Given any integer A and positive integer B: 1) Q and R exist 2) Q and R are unique
Proof that Q and R exist
(One approach is to use the Well Ordering Principle of Integers, but I will use an approach that is arguably simpler) Suppose we have an integer A and a positive integer B. Given any real number A and any positive real number B if I divide A by B, I will have an integer part (before the decimal),w, and a fractional part,f, after the decimal.
If A is >= 0 then w and f are non-negative: A/B = w + f where 0 ≤ f < 1 A = B * w + B * f w is an integer (it is the whole part) we can simply label it as Q, i.e. Q = w
by multiplying 0 ≤ f < 1 by B we find that B * 0 ≤ B * f < B * 1 0 ≤ B * f < B we can simply label the term B * f as R i.e. R= B * f We have shown that 0 ≤ R < B To show that R is an integer we can say that: A = B * w + B * f A = B * Q + R (using our new labels) we can rearrange this to: R = A - B * Q but A, B, Q are integers and the result of any integer minus the product of integers is still an integer (this a property of integers) so R must be an integer
so we have shown that given any integer A >= 0 and any positive integer B, there exist integers Q and R such that: A= B * Q + R where 0 ≤ R < B
if A is negative then A/B = w + f where -1 < f ≤ 0 A = B * w + B * f
if f = 0 then label w as Q and B * f as R A = B * Q + R since B * f = R = 0 we can say that R satisfies 0 ≤ R < B and that R is an integer
if -1 < f < 0 A = B * w + B * f A = B * ( w - 1) + B * (f + 1) label w-1 as Q, which is an integer
add 1 to -1 < f < 0 0 < f + 1 < 1 multiply by B 0 < B * ( f + 1 ) < B we can simply label the term B * (f + 1) as R We have shown that 0 < R < B which satisfies 0 ≤ R < B
To show that R is an integer, in this case, we can say that: A = B * ( w - 1) + B * (f + 1) A = B * Q + R (using our new labels) we can rearrange this to: R = A - B * Q but A, B, Q are integers and the result of any integer minus the product of integers is still an integer (this a property of integers) so R must be an integer
so we have shown that given any integer A < 0 and any positive integer B, there exist integers Q and R such that: A= B * Q + R where 0 ≤ R < B
so we have shown that given any integer A and any positive integer B, there exist integers Q and R such that: A= B * Q + R where 0 ≤ R < B
Proof that Q and R are unique
Suppose we have an integer A and a positive integer B. We have shown before that Q and R exist above. So we can find at least one pair of integers, Q1 and R1, that satisfy A= B * Q1 + R1 where 0 ≤ R1 < B And we can find at least one pair of integers, Q2 and R2, that satisfy A= B * Q2 + R2 where 0 ≤ R2 < B
For labeling purposes, R2 is >= than R1 (if not we could just switch the integer pairs around). We will show that Q1 must equal Q2 and R1 must equal R2 i.e. Q and R are unique
We can set the equations equal to each other B * Q1 + R1 = B * Q2 + R2 B * (Q1 - Q2) = (R2 - R1) (Q1 - Q2) = (R2 - R1)/ B
since R2 >= R1 we know that R2 - R1 is >= 0 since R2 <B and R1 >= 0 we know that R2-R1 < B So we can say that 0 ≤ R2 - R1 < B divide by B 0 ≤ (R2 - R1)/B < 1 but from above we showed that (Q1 - Q2) = (R2 - R1)/ B and Q1 - Q2 must be an integer since an integer minus an integer is an integer so (R2 - R1)/B must be an integer but its value is >= 0 and < 1. The only integer in that range is 0. So (R2- R1)/B= 0 ,thus R2-R1 =0 ,thus R2 = R1 also Q1-Q2 = 0 thus Q1 = Q2
Thus we have shown that Q1 must equal Q2 and R1 must equal R2 i.e. Q and R are unique
When we want to prove some properties about modular arithmetic
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
https://en.wikipedia.org › wiki › Modular_arithmetic
we often make use of the quotient remainder theorem. It is a simple idea that comes directly from long division. We can see that this comes directly from long division. When we divide A by B in long division, Q is the quotient and R is the remainder.
The quotient-remainder theorem says that when any integer n is divided by any pos- itive integer d, the result is a quotient q and a nonnegative remainder r that is smaller than d. n = dq + r and 0 ≤ r < d.
The remainder theorem states that when a polynomial p(x) is divided by (x - a), then the remainder = f(a). This can be proved by Euclid's Division Lemma. By using this, if q(x) is the quotient and 'r' is the remainder, then p(x) = q(x) (x - a) + r.
Remainder Theorem is an approach of Euclidean division of polynomials. According to this theorem, if we divide a polynomial P(x) by a factor ( x – a); that isn't essentially an element of the polynomial; you will find a smaller polynomial along with a remainder.
The Remainder Theorem tells us that, in order to evaluate a polynomial p(x) at some number x = a, we can instead divide by the linear expression x − a. The remainder, r(a), gives the value of the polyomial at x = a.
The dividend divisor quotient remainder formula can be applied if we know either the dividend or remainder or divisor. The formula can be applied accordingly. For dividend, the formula is: Dividend = Divisor × Quotient + Remainder. For divisor, the formula is: Dividend/Divisor = Quotient + Remainder/Divisor.
Solution: Consider the value of a to be 4. Therefore, f ( 4 ) = 0. Remainder Theorem is a way of addressing Euclidean's division of polynomials. It states that when a polynomial is p(a) is divided by another binomial (a – x), then the remainder of the end result that is obtained is p(x).
The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , x - a, the remainder of that division will be equivalent to f(a).
It states that when a polynomial P(x) is divided by a linear polynomial x – a, the remainder is equivalent to P(a). This theorem simplifies polynomial evaluation by allowing one to find the remainder of a division by directly substituting a value into the polynomial.
The remainder theorem can be used when you want to find the remainder of a polynomial division. In other words, it can be used to determine the remainder when a polynomial is divided by another polynomial. The remainder theorem is also sometimes referred to as the polynomial remainder theorem.
The reminder theorem is only true when the divisor is a linear polynomial. That means it cannot be utilized when the divisor is something else and if the degree of the divisor polynomial is more than 1 , the sole way to find the remainder is polynomial long division.
Answer: According to the remainder theorem, if polynomial f(x) is divided by a linear binomial of the variable (x – a) then the remainder will be f(a). On the other hand, factor theorem states that if f(a) = 0 in this case then the binomial (x -a) is a factor of polynomial f(x).
The remainder theorem says that when dividing a polynomial f(x) by x - a, the remainder will be equal to f(a). The factor theorem says that if f(a) = 0 then x - a is a factor of the polynomial f(x). Putting these two theorems together tells us that if the remainder is 0 then we have found a factor of the polynomial.
A remainder is always less than the divisor. If one number (divisor) divides the other number (dividend) completely, then the remainder is 0. It is referred to as a complete division. If a dividend is a multiple of the divisor, then the remainder is 0.
The quotient is the number of times a division is completed fully, while the remainder is the amount left that doesn't entirely go into the divisor. For example, 127 divided by 3 is 42 R 1, so 42 is the quotient, and 1 is the remainder.
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two ...
The formula for Quotient = Dividend ÷ Divisor. 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.
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