CAT Basic remainder theorem and its application to determine the remainder when a large number is divided by small numbers - Practice Questions & MCQ

Introduction:

The remainder theorem states that if a polynomial f(x) is divided by x - a, then the remainder is f(a). While the theorem was originally designed for polynomials, it has wide applications in the world of number theory, especially when dealing with big numbers.

Application:

To find the remainder when a large number is divided by a small number, you can use properties of numbers and certain tricks to simplify the problem. For example, when finding the remainder of a number with the last digit 5 when divided by 4, you can only consider the last two digits.

Foundation Building Questions:

Question: 

Find the remainder when 678912345 is divided by 8.

Solution: 

Notice that powers of 9 cycle every 2 powers with respect to a modulus of 8:

And so on...

Given the power is odd (12345), the remainder will be  mod 8 which is 1.

But we must also consider the units digit 9. Therefore, the actual computation should be:

So, the remainder is 1.

Tips and Tricks:

1. Patterns: Look for patterns in the powers of the base number with respect to the divisor. Many numbers show a cyclical pattern.

2. Last few digits: When dividing by small numbers like 2, 4, 5, 8, 25, or 125, consider only the last 1, 2, or 3 digits of the dividend.

3. Modular Arithmetic: Master modular arithmetic. It can greatly simplify problems related to remainders.