Negative Remainder Theorem and its Application

Negative Remainder Theorem

First of all you need to understand what are positive remainder and negative remainder?
Say, a number 37 is divided by 7 then what would be the remainder?
Yes, 2 is the actual remainder which is also a positive remainder. It also gives a negative remainder. The negative remainder would be -5. See, the positive remainder is 2 which are just more than 35 and -5 is just less than 42. Notice, 35 and 42 is divisible by 7.

Click Here to read Positive Remainder Theorem.

Considering a question:
Find the remainder when $$\frac{{14 \times 15}}{8}.$$

Solution:
$$\frac{{14 \times 15}}{8}$$  ==R==> $$\frac{{6 \times 7}}{8}$$  ==R==> 2 [by positive remainder theorem]
$$\frac{{14 \times 15}}{8}$$  ==R==> $$\frac{{\left( { - 2} \right) \times \left( { - 1} \right)}}{8}$$   ==R==> 2 (by negative remainder theorem.)

If the answer comes out negative,
$$\frac{{62 \times 63 \times 64}}{{66}}$$   ==R==> $$\left( { - 4} \right) \times \left( { - 3} \right) \times \left( { - 2} \right)$$    ==R==> -24.
But a remainder never be negative, then remainder is given by (66 - 24) = 42.

You can use positive and negative remainder theorem both at same time as well. This depends on your convenience.
$$\frac{{243 \times 245 \times 247 \times 249}}{{12}}$$    ==R==> $$3 \times 5 \times \left( { - 5} \right) \times \left( { - 3} \right)$$    ==R==> $$\frac{{15 \times 15}}{{12}}$$  ==R==> 9.

Here, we have used both the theorem same time as our convenient.

Click Here for Solved Examples on Number System