# Playing with Numbers

Numbers are the mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, “5” is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number.

## Types of Numbers

The important types of numbers are:

**Natural Numbers****Whole Numbers****Integers****Rational Numbers.**

**Natural Numbers**

Natural Numbers are the basic numbers in Number System, and they are used in our daily life for counting purposes. For this reason, they are also called “**Counting Numbers**“.

- The Range of Natural Numbers: 1 to ∞
- Generally, Natural Numbers are represented by
**N**. - Natural Numbers do not contain 0 and negative numbers.

For example:

1, 2, 3, 4, 5, 6, 7…. are natural numbers

**Whole Numbers**

Whole numbers are the Natural Numbers including zero(0).

- Generally, Natural Numbers are represented by
**W**. - Range of Whole Numbers: 0 to infinite.

For example:

0, 1, 2, 3, 4, 5, 6, 7….. are Whole numbers

**Integers**

Integers are the numbers including positive numbers, negative numbers, and zero.

- Integers are represented by
**Z**. - Range of Integers: -ve Infinite to +ve Infinite, including 0.

For example:

…., -3, -2, -1, 0, 1, 2, 3, ….. are Integers

**Rational Numbers**

Rational Numbers are the numbers that are expressed in Fraction form (like 2/3, 1/4).

- Rational numbers can be either positive or negative.
- All fractions are rational numbers but vice-versa is not true.

For example:

-1/2, 1/2, 3/4….. are Rational Numbers

## Numbers in General Form

The general form of a two-digit number is: **ab=(10 × a)+b**

Here ab is the usual form and (10 × a)+b is the generalized form of the two-digit number.

For example:

- 48 = 10 × 4 + 8
- 67 = 10 × 6 + 7

The general form of a three-digit number is: **abc =(a × 100) + (b × 10) + (c × 1)**

Here abc is the usual form of the number and (a × 100) + (b × 10) + (c × 1) is the generalized form of the 3-digit number.

For example:

- 237 = 2 × 100 + 3 × 10 + 7 × 1
- 437= 4 × 100 + 3 × 10 + 7 × 1

Therefore, in order to express the number in its general form, the units place digit should be multiplied by 1, tens place digit should be multiplied by 10, Hundred place digits should be multiplied by 100, and so on. Finally, all these multiplied values should be added together. It can be understood with the below example:

**Example 1:**

Given Number = 52

Units place digit = 2

Tens place digit = 5

Multiply units place digit with 1 (i.e, 2 x 1)

Multiply Tens place digit with 10 (i.e, 5 x 10) and finally add them together.

52 = (5 x 10) + (2 x 1)

**Example 2:**

Given Number = 351

Units place digit = 1

Tens place digit = 5

Hundreds Place digit = 3

Multiply units place digit with 1 (i.e, 1 x 1)

Multiply Tens place digit with 10 (i.e, 5 x 10)

Multiply Hundreds place digit with 100 (i.e, 3 x 100) and finally add them together.

351 = (3 x 100) + (5 x 10) + (1 x 1).

## Game With Numbers

Games with numbers are the tricks that are applicable to every two digits and three-digit number.

### Reverse the 2-digit numbers and add them

Choose any 2-digit number and reverse it, now add both the reversed number and original number then the resulting number will be divisible by 11 and also quotient will be equal to the sum of digits.

Example 1:Consider 2-digit number 62

Reverse of the number 26

Add both of them = 62 + 26 = 88

88 is divisible by 11.

When 88 is divided by 11 it gives quotient 8 (which is equal to sum of digits in 62 which is 6 + 2)

.

Example 2:Consider 2 digit number 33

Reverse of the number 33

Add both of them = 33 + 33 = 66

66 is divisible by 11.

When 66 is divided by 11 it gives quotient 6 (which is equal to sum of digits in 33 which is 3 + 3).

### Reverse the 3-digit numbers and Subtract them

Choose any 3-digit number and reverse it, now subtract both the reversed number and original number (Always calculate the absolute difference i.e, the difference should be greater than 0) then the resulting number will be divisible by 99 and also quotient will be equal to the absolute difference between 1st and 3rd digit of the given number.

Example 1:Consider 3-digit number 734

Reverse 3-digit number 437

Absolute difference = 734 – 437 = 297

297 is perfectly divisible by 99.

When 297 is divided by 99 it gives quotient 3 (which is equal to difference between 1st and 3rd digit in given number = 7 – 4 = 3).

Example 2:Consider 3-digit number 162

Reverse 3-digit number 261

Absolute difference = 261 – 162= 99

99 is perfectly divisible by 99.

When 99 is divided by 99 it gives quotient 1 (which is equal to difference between 1st and 3rd digit in given number = 2 – 1 = 1).

### Forming 3-digit numbers with given three-digits

Choose any 3-digit number (let’s say “XYZ” is a 3-digit number we chose). Now generate the following two numbers as follows:

- Shift the one’s digit to the leftmost end (i.eXYZ —> ZXY).
- Shift the Hundredths digit to the rightmost end (i.e XYZ —> YZX).

Now add all these 3 numbers (XYZ + ZXY + YZX). This result will be perfectly divisible by 37.

Example 1:Consider 3-digit number 162

Shift one’s digit to the leftmost end = 216

Shift the Hundreds digit to the rightmost end = 621

Now add all these 3 numbers = 162 + 216 + 621 = 999

999 is perfectly divisible by 37 (Quotient 27).

Example 2:Consider 3 digit number 163

Shift ones digit to the leftmost end = 316

Shift the Hundreds digit to the rightmost end = 631

Now add all these 3 numbers = 163 + 316 + 631 = 1,110

1,110is perfectly divisble by 37 (Quotient 30).

### Letters for Digits

What number multiplied by 2 gives 60? What is that number? The number is 30. That number is not known; assume that number as a variable and what value to be assigned to that variable in order to get the result as 60. Variable is defined as an unknown quantity. No definite value and it keeps changing. It is a puzzle-solving type in which a value for a specified letter needed to be determined. The value of a letter has to have only one digit. If there are multiple letters in a puzzle same values cannot be assigned to multiple letters. The value to a letter cannot be zero if the letter is in the starting position.

For example:3c + 45 = k4

Now, What are the values of c and k?

Put c =9 and k = 8

we get,

39 + 45 = 84

Hence, it is a puzzle game in which some digits in the Arithmetic Sum were replaced by Alphabetical letters and the task is to find out the actual digits. It is just like cracking a code.

More such interesting examples are solved below. It’s not only fun and enjoyable but also its sharpness our brain by solving such problems.

Example 1:1 X 1

3 9 X

– – – (+)

4 9 1

– – –

Here the task is to find the value of X.

Consider units digit column –> 1 + X = 1

–> X = 0 (clearly only 0 can satisfy this equation).

Consider Tens digit column –> X + 9 = 9

–> X = 0 (clearly only 0 can satisfy this equation).

The value of X is 0.

Example 2:4 X 2

3 6 X

– – – (+)

7 8 4

– – –

Here the task is to find the value of X.

Consider units digit column –> 2 + X = 4

–> X = 2

Consider Tens digit column –> X + 6 = 8

–> X = 2 (clearly only 2 can satisfy this equation).

The value of X is 2.