Understanding of Remainder theorem

Remainder Theorem

There is two types of remainder theorem.Positive Remainder Theorem and Negative Remainder Theorem. Here, we will understand the basics of positive remainder theorem.

It can be best expressed by an example.
Remainder of,
1421 × 1423 × 1425 when divided by 12 can be given as:
$$\frac{{1421 \times 1423 \times 1425}}{{12}}$$     ==R==> $$\frac{{5 \times 7 \times 9}}{{12}}$$
[we have taken individual remainders of the product. R = Remainder.]

$$\frac{{5 \times 7 \times 9}}{{12}}$$   ==>R==> $$\frac{{35 \times 9}}{{12}}$$  ==R==> $$\frac{{11 \times 9}}{{12}}$$  ==R==> gives us a remainder 3.

In above example,
we have used series of remainder theorem transformation (denoted by ==R==>) and equality transformations to transform a difficult looking expression in to a simple one.

Let’s take few more examples to understand it better.
Remainder when
$$\frac{{17 \times 23 \times 126 \times 38}}{8}$$
$$\frac{{17 \times 23 \times 126 \times 38}}{8}$$     ==R==> $$\frac{{1 \times 7 \times 6 \times 6}}{8}$$    ==R==> $$\frac{{42 \times 6}}{8}$$  ==R==> $$\frac{{2 \times 6}}{8}$$  ==R==> 4.
This gives remainder 4.

Remainder when $$\frac{{37 \times 43 \times 51}}{7} + $$   $$\frac{{137 \times 143 \times 151}}{9}$$

Given expression contains two parts. First part is $$\frac{{37 \times 43 \times 51}}{7}$$    and second one is $$\frac{{137 \times 143 \times 151}}{9}.$$

First we take individual remainder of first part of expression.
$$\frac{{37 \times 43 \times 51}}{7}$$    ==R==> $$\frac{{2 \times 1 \times 2}}{7}$$   ==R==> $$\frac{4}{7}$$.
$$\frac{{137 \times 143 \times 151}}{9}$$    ==R==> $$\frac{{2 \times 8 \times 6}}{9}$$   ==R==> $$\frac{{2 \times 48}}{9}$$  ==R==> $$\frac{6}{9}$$.

Now, $$\frac{4}{7} + \frac{6}{9} = \frac{{36 + 42}}{{63}}$$    ==R==> $$\frac{{78}}{{63}}$$ ==R==> 15.

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