Remainder Theorem

The remainder theorem is a result often employed in the calculus of one variable. It provides a systematic way to find out whether an integer c exists that satisfies  x = ax + c, for all x in R, then remainder theorem tells us that there is a unique integer c such that

x – ax = c.

The remainder theorem was introduced by the great French mathematician Pierre de Fermat in 1665. He used it to show that every integer is the remains of a perfect (or Fermat) integer multiplied by itself. It was used as a way of finding modular multiplicative functions.

What is the remainder theorem?

Let’s understand the remainder theorem. 

The remainder theorem states that:

If x = ax + c, then a unique integer c exists such that

x – ax = c.

This statement can always be verified by subtracting the common multiple of x and a from both sides of the equation. If it does not work out, we can potentially deal with an undefined expression.

We usually apply the remainder theorem to find c when looking for a common factorization polynomial of x and a. He used it to show that every positive integer is the remains of a perfect (or Fermat) integer multiplied by itself.

The Remainder Formula:

The above statement of the remainder theorem can be mathematically expressed as follows:

x – ax = (x – b) mod c, where b is a multiple of c.

Now, this may look more complicated than it is. Think of it like this:  x – ax is the same as x mod a + ax mod c for some integer b.

The remainder theorem formula is a well-known formula employed in number theory, modular arithmetic, and cryptography. Notice that the remainder equation tells us how to get from x – ax = c to x mod a + ax mod c.

What is the Chinese Remainder Theorem? 

This theorem is known as the Chinese remainder theorem for its connection to Chinese mathematics.

The theorem can be mathematically stated as follows:

x – ax + bx – cx = d mod p, where p is a prime and b and c are integers.

Notice that the last statement of the theorem is different from that of the remainder theorem. It can be seen by examining each equation x – ax = c and x – cx = d to have solutions.

The first equation tells us that it is possible to solve for x when dividing both sides of the equation by c. The second equation says that we can solve for x if we know a, b and d. Both solutions are unique as long as a, b, and d are unique.

The Chinese remainder theorem can be used to find solutions in cases where the remainder theorem cannot be applied.

Proof of Remainder Theorem

The remainder theorem can be proved by induction as follows:

If x = ax + c, it is obvious that there exists a unique integer c such that

x – ax = c.

Let us now suppose that the theorem is true for all x such that |x| ≤ k and consider x  ≥ k + 1.

It means that a unique integer c exists such that x – ax = c, which means that |x| – 1 is a multiple of c. It is also obvious that  |x| − 1 and |x – 1| are also multiples of c, which means we can apply the remainder theorem again to get

x − 1 = (x − b) mod c.

Application of Remainder Theorem and Chinese Remainder Theorem

It is often easier to apply the Chinese remainder theorem to fit a particular case.

Case 1: The remainders theorem applied: If |x| is an integer such that |x| ≤ k, then there exists a unique integer c such that x – ax = c. We have just shown this in an inductive proof.

Case 2: When the remainders theorem cannot be applied: Suppose we are given x  ≥ k + 1. Then we can apply the Chinese remainder theorem to get

(x − b) mod c = (x − a) mod c.

We know that |x| ≥ k + 1, which means that |x| – a is an integer.

This means that

(x − b) mod c = (|x| − a) mod c = |(k + 1)| mod c, which is exactly what we seek.

The remainder theorem can be used to solve equations that can be expressed in terms of polynomial factors. 

Summary of the Remainder Theorem

The remainder theorem can be used to find a common factorization polynomial of a rational number and an integer if it exists. Use the Chinese remainder theorem whenever possible when proving results that cannot be proven using the remainder theorem.

Conclusion

The remainder theorem allows us to solve problems involving polynomials that cannot be easily solved using simple arithmetic. It is an example of a theorem that shows how advanced mathematics can simplify and speed up tasks in other areas of study.

In both cases, i.e., in the remainder theorem and the Chinese remainder theorem, we can prove that there are integers that satisfy x = ax + c for all x in R.

The main difference is how we go about solving this problem.

 If x − ax and |x| – 1 are integers, then so too is c.