We use numbers in our day-to-day life. We buy everything with money and measure its quantity with the help of Numbers only. Therefore Numbers play a very significant role in our life.

**The common Representation of a Number is as follows: **

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.

- 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.

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

## Game With Numbers

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

### 1. Reversing Two – Digit Number

To Reverse a two-digit number we have to interchange the digits ie 35 when Reversed 53, ones digit place to tens digit place and tens digit place to ones digit place. Now add them up and divide the result obtained by 11 we observe that there is no remainder left.

By adding we get 88 by dividing by 11 the remainder is 0.

### 2. Reversing Three – Digit Number

Let us consider a three-digit number 345 on reversing it we get 543. Subtract the larger number from the smaller number and divide the result obtained by 99. We get the remainder as 0.

543 – 345 = 198, we get quotient as 2 when divided by 99.

### 3.Forming a Three Digit Number From The Given Three Digits

If the three digits are given we can generate 6 combinations of numbers

Eg: 3,7,8 the numbers can be generated using three digits as follows 378, 387, 873, 837, 783, 738.

Let us think of a number ie 754

Form a number such that the last digit of a number should become first digit. Not changing the remaining digits. Ie 7 and 5.

We get 475 and form a number from the number we get, shift the last digit to the place of the first digit and the number we get is 547.now sum up the original numbers along with newly formed numbers and divide the result by 37. We get the remainder as 0.

**Letter For Digits**

2 is multiplied by a number = 60. What is that number ? The number is 30. We don’t know that number; we 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 we have to find a value for a specified letter. The value to 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. Eg: 3c + 45 = k4 here what are the values of c and k?. c =9 and k = 8

Putting values of c and k we get, 39 + 45 = 84

9+5=14,1 carry last digit is 4 by substituting 9 as a value to the letter or a variable c we get the last digit as 4.

Let us consider an example A0 x B0=1500 what are the values of A, B? A =3 OR A = 5 B = 5 0R B = 3

Its very simple by looking at this we can determine the values of A and B

i.e 50 × 30 or 30 × 50.

You can try more such interesting examples and try it. It’s not only fun and enjoyable but also its sharpness our brain by solving such problems.

## Introduction To Divisibility Rule

Suppose I have 537 chocolates and I have to distribute them among my 9 friends. How can I do it? Dividing 537 by 9 and I have left with some chocolates(remainder) that means 537 is not divisible by 9 exactly. Dividing is simple to check that the number is exactly divided by the divisor ie Remainder is 0 or not when we have 2 or 3-digit numbers. If the number is too large and it takes a long time to perform the actual division. How can we know that a number is divisible by a particular divisor or not. Here comes the concept of divisibility rules: quick easy and simplest way to find out the divisibility of a number by a particular divisor.

**Divisibility Rule For 2**

A number is divisible by 2 if the last digit of the number is any of the following digits 0, 2, 4, 6, 8. The numbers with the last digits 0, 2, 4, 6, 8 are called even numbers, Eg: 2580, 4564, 90032 etc. which are divisible by 2.

### Divisibility Rules For 3 And 9

A number is divisible by 3 if the sum of its digits is divisible by 3. Eg: 90453 (9 + 0 + 4 +5 + 3 = 21) 21 is divisible by 3. 21 = 3 × 7. Therefore, 90453 is also divisible by 3. Same rule is also applicable to test whether the number is divisible by 9 or not but the sum of the digits of the number should divisible by 9 in the above example 90453 when we add the digits we get the result as 21 which is not divisible by 9. Eg: 909, 5085, 8199, 9369 etc are divisible by 9. Consider 909 (9 + 0 + 9 = 18). 18 is divisible by 9(18 = 9 × 2). Therefore, 909 is also divisible by 9.

A number which is divisible by 9 also divisible by 3 but a number that is divisible by 3 not have surety that it is divisible by 9.

Eg: 18 is divisible by both 3 and 9 but 51 is divisible only by 3, can’t be divisible by 9.

### Divisibility Rules For 5 And10

A number is divisible by 5 if that last digit of that number is either 0 or 5. Eg:500985

3456780, 9005643210, 12345678905 etc.

A number is divisible by 10 if it has only 0 as its last digit. Eg: 89540, 3456780, 934260 etc. A number which is divisible by 10 is divisible by 5 but a number which is divisible by 5 may or may not be divisible by 10.10 is divisible by both 5 and 10 but 55 is divisible only by 5 not by 10.

### Divisibility Rules For 4 , 6 And 8

A number is divisible by 6 if it is divisible by both 2 and 3.

A number is divisible by 4 if the last two digits are divisible by 4. Eg: 456832960 In this example the last two digits are 60 that are divisible by 4 i.e.15 × 4 = 60. Therefore, the total number is divisible by 4.

Considering the same example let us check the divisibility rule for 8. If a number is divisible by 8 its last three digits should be divisible by 8 i.e. 960 which is divisible by 8 therefore the total number is divisible by 8.

### Divisibility Rule For 11 And7

Consider the above number that we used to test the divisibility with 4 and 8

456832960 mark the even place values and odd place values. Sum up the digits in even place values together and sum up the digits in odd place values together.

Digits | Place Value |

4 | 0 |

5 | 1 |

6 | 2 |

8 | 3 |

3 | 4 |

2 | 5 |

9 | 6 |

6 | 7 |

0 | 8 |

Now we have to sum up the digits in even place values ie 0 + 2 + 4 + 6 + 8 = 4 + 6 + 3 + 9 + 0 = 22

Now we have to sum up the digits in odd place values ie

1+ 3 + 5 + 7 = 5 + 8 + 2 + 6 = 21

Now calculate the difference between the sum of digits in even place values and the sum of digits in odd place values if the difference is divisible by 11 the complete number ie 456832960 is divisible by 11

Here the difference is 1, (22-21) it is divisible by 11. Therefore, 456832960 is divisible by 11.

Consider the number 5497555 to test if it is divisible by 7 or not.

Add the last two digits to the twice of the remaining number repeat the same process until it reduces to a two-digit number if the result obtained is divisible by 7 the number is divisible by 7.

55 + 2(54975) = 109950 + 55 = 110005

05 + 2(1100) = 2200 + 05 = 2205

05 + 2(22) = 44 + 5 = 49

Reduced to two-digit number 49 which is divisible by 7 i.e 49 = 7 × 7

Therefore, 5497555 is divisible by 7.

*Some More Divisibility Rules*

Co-primes are the pair of numbers that have 1 as the common factor. If the number is divisible by such co-primes the number is also divisible by product of the co-primes. Eg: 80 is divisible by both 4 and 5 they are co-primes have only 1 as the common factor so the number is also divisible by 20 the product of 4 and 5

21 = 3 × 7

12 = 3 × 4

22 =11 × 2

14 = 2 × 7

15 = 3 × 5

30 = 3 × 10

18 = 2 × 9

28 = 4 × 7

26 =13 × 2.

If a number is divisible by some numbers say X that number is also divisible by factors of x. Eg: if a number is divisible by 40 then it is divisible by its factors Ie: 5, 10, 2, 4, 8, 20.

### Divisibility Rule For 13

* *If a number to be divisible by 13 add 4 times the last digit of the number to the rest of the number repeat this process until the number becomes two digits if the result is divisible by 13 then the original number is divisible by 13. Eg: 333957

(4 × 7) + 33395 = 33423

(4 × 3) + 3342 = 3354

(4 × 4) + 335 = 351

(1 × 4) + 35 = 39

Reduced to two-digit number 39 is divisible by 13 Therefore 33957 is divisible by 13.

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