## Number Theory

### 1. Introduction

Numbers are used to measure and count. The number system which is globally in use is the decimal number system, where the base is $10$. This just means that there are $10$ digits in this number system. They are $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$. Unless specified otherwise, the**base 10 number system**will be used for all calculations.

The base $9$ number system will use the first $9$ digits mentioned above which are $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$ and $8$. Likewise, the base $8$ number system will not include the digits $8$ and $9$. So, the base $2$ number system will include only the digits $0$ and $1$ - called the binary number system.

As number systems of base higher than $10$ have never been widely used, additional digits used have widely varied. The widely agreed form of hexadecimal (base $16$) representation is to use the letters A, B, C, D, E and F as the digits after $9$. In the management exams, typically these letters in that order have been used to represent digits for number systems with base higher than $10$. We will discuss these in detail in the $Number \space Systems$ lesson.

#### 1.1 Decimal Number System

A number in the decimal number system, for example $94.25$,$94.25 = 9 \times 10^1 + 4 \times 10^0 + 2 \times 10^-1 + 5 \times 10^-2$

$94.25 = 9 \times 10 + 4 \times 1 + \dfrac{2}{10} + \dfrac{5}{100}$

Note that digits to the left of the decimal point are multiplied with $10^0$, $10^1$, $10^2$ and so on, while the digits to the right of the decimal point are multiplies with $10^{-1}$, $10^{-2}$, and so on, in that order.

Therefore, digits in the places to the left of the decimal point are units digit, tens digit, hundreds digit, thousands digit and so on. The digits in places to the right of the decimal point are called tenths digit, hundredths digit, thousandths digit and so on.

In the above example, $9$ is tens digit, $4$ is the units digit, $2$ is the tenths digit and $5$ is the hundredths digit.

#### 1.2 Application in MBA tests

Questions in MBA tests might require us to find $2$-digit or $3$-digit numbers with certain conditions around the digits. In these questions, we assume the digits to be variables and form the numbers.A $2$-digit number can be written as $10a + b$, where $a$ and $b$ are the tens and units digit respectively.

Likewise, a $3$-digit number can be written as $100a + 10b + c$ where $a$, $b$ and $c$ are the hundreds, tens and units digit respectively.

Or, a $4$-digit numbers can be written $1000a + 100b + 10c + d$ , where $a$, $b$, $c$ and $d$ are the thousands, hundreds, tens and units digit respectively.

### Example 1

When a $3$-digit number is reversed, we get another $3$-digit numbers. If the difference between these two numbers is $297$, then what is the difference between the hundreds digit and the units digit of the number?

When the digits are reversed, the number formed is $\bm{100c} + \bm{10b} + \bm{a}$.

$100 a + 10 b + c - (100 c + 10 b + a) = 297$

$\implies 99 a - 99c = 297$

$\implies a - c = 3$

### Solution

Let $x$, $y$ and $z$ be the hundreds, tens and units digit of the number. Therefore, the number is $\bm{100a} + \bm{10b} + \bm{c}$.When the digits are reversed, the number formed is $\bm{100c} + \bm{10b} + \bm{a}$.

$100 a + 10 b + c - (100 c + 10 b + a) = 297$

$\implies 99 a - 99c = 297$

$\implies a - c = 3$

**Answer**: $3$