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Understanding of Remainder theorem through divisibility property

divisibility and remainder theorem

First thing you should keep in mind is that in theory of numbers we’ll mostly be talking about numbers from the set of whole or the set of real numbers. One of the most important basics in theory of number is the definition of divisibility and some basic rules.

divisibility and remainder theorem

Definition 1. If there exists  such that b = d * a, we say that a divides b, where a and b are whole numbers and a \not= 0. It is said that b is the multiple of a, and a is the divisor of b. If a divides b we denote it as a \mid b and if not a \phi b.

 

Proposition 1. If  a \mid b and b \not= 0 then \mid a \mid \le \mid b \mid.

Proof. If a \mid b and b \not= 0 \Rightarrow (by the Definition 1.) \ni d such that b = d * a, d \not= 0, \mid d \mid \ge 0.
Since we know that b = d * a we can affect this equation with absolute value. We get that \mid b \mid = \mid d \mid * \mid a \mid, and we got where we wanted to be.
We wanted to prove that \mid a \mid \le \mid b \mid and we got that \mid b \mid is equal to the product of some positive whole number d and \mid a \mid which proves our proposition.

 

Proposition 2. If a \mid b, then a divides every multiple of b.

Proof. Let’s say that b = a * d_1 and that c is the multiple of b. According to that, there exists number d_2 such that c = b * d_2 and since b = a * d_1 we have that c = a * d_1 * d_2. Since d_1 and d_2 are both whole numbers, so is their product and we got that the multiple of b-c, is the product of some whole number and a. This means that a \mid c.

 

Proposition 3. If a \mid b and a \mid c, then a \mid (b \pm c) and a \mid (b * c).

Proof. If a \mid b and a \mid c then exist d_1 and d_2 such that b = a * d_1 and c = a * d_2. Since we want to prove that a divides b \pm c we’ll simply include b and c into this expression. This means that we’ll have b \pm c = a (d_1 \pm d_2). Since d_1 and d_2 are whole number, so are their sum and difference and we got that b \pm c is equal to the product of number  with some whole number. This means that a \mid (b \pm c).

The same goes for the product of b and c:
b * c = a^2 (d_1 * d_2) = a (a * d_1 * d_2)
and in the same way we got that b * c is equal to the product of number a and some whole number which means that a \mid (b * c).

 

This relation – “to be divisible” is relation of partial order. This means that are fulfilled following three conditions:
1. Reflexive property – This means that a certain number is always divisible by itself or: a \mid a

2. Antisymmetric property – This means that if one number is divisible by another number, and in the same time that other number is divisible by the first number, those two numbers must be equal. a \mid b and b \mid a \Rightarrow a = b

3. Transitive property – it tells us that if one number divides second number which divides third number, then the first number will also divide third number. a \mid b and b \mid c \Rightarrow a \mid c

 

The remainder theorem

For real number a and whole number b exist unique whole numbers q and r such that

b = a * q + r, where 0 \le r \le a.

Proof. We’ll divide this proof into two parts. First part is in which we will prove existence and second in which we will prove uniqueness.
1. Existence: let’s observe rational number \frac{b}{a} and define number q such that q \le \frac{b}{a} \le q + 1

If we subtract q from these inequalities we’ll get: 0 \le \frac{b}{a} - q < 1
Now we can define number r: r = a (\frac{b}{a} - q) = b - aq
This means that b = r + aq

If we multiply the inequality 0 \le \frac{b}{a} - q < 1 with a we get: 0 \le b - aq < a
Since we defined r as r = b - aq we now have 0 \le r < a which proves existence.

2. Uniqueness: here we’ll assume there exist whole numbers d_1 and d_2 such that aq_1 + r_1, 0 \le r < a
This would mean that aq + r = aq_1 + r_1.
a(q - q_1) = r_1 - r. If we assume that r_1 \ not= r, then so is q \not= q_1.
If r \not= r_1 \Rightarrow 0 < \mid r_1 - r \mid < a.

From a(q - q_1) = r_1 - r (by applying absolute value) we get that
\mid r_1 - r \mid = \mid a \mid * \mid q - q_1 \mid \ge \mid a \mid = a which is in contradiction with the statement that \mid r_1 - r \mid < a.
This means that r_1 = r if and only if q = q_1. We say that r is the remainder in division of number b with number a, and q is the quotient of integer division - q= \frac{b}{a}.

Polynomials

The Remainder Theorem for polynomials an application of Euclidean division on polynomials. Let’s explain it closer…

When we divide any polynomial f(x) by a linear polynomial (x - c) we get another polynomial:

f (x) = (x - c) q(x) + r

The point of this theorem is realizing what this remainder r really is. Let’s try to replace x with c and see what happens.

f(c) = (c - x) q(c) + r

f(c) = 0 * q(c) + r

f(c) = r

When we divide a polynomial f(x) by a linear polynomial (x - c), where c \in \mathbb{R}, the remainder r is equal to f(c).

This theorem is very useful, especially if we want to know whether one polynomial is divisible by another, linear polynomial. If the remainder is equal to zero, then he is divisible, if it is any other number, he is not.

Example 5. Is a polynomial divisible by f(x) = 5x^4 + 3x - 6x + 3 divisible by (x + 2).

We have a linear polynomial (x + 2) which means that our c is equal to – 2. Now we calculate f(-2) = 0.

f(-2) = 5 * (- 2) + 3 * (- 2) - 6 * (- 2) + 3 = - 10 - 6 + 12 + 3 = - 1

- 1 \not= 0 which means that polynomial f(x) is not divisible by (x + 2).

 

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