MULTIPLY NUMBERS FROM 101 THROUGH 109
      In this case the examples would be:-
         101x101
         104x107
         1 05 x105
        106x108
        109x109
And so on----
The total examples will be 45.
Procedure:-
•  The answer will be a five digit number beginning with 1.
•  The next two digits of the answer will equal the sum of ones digits.
•  The last two digits of the answer will equal the product of the ones digits.
Let’s use the trick.
Consider the problem: 102x107.
Step1:- Begin the answer with 1 ---.>   1
Step 2:-Add the ones digits    ------>2+7= 09.
Step 3:- Multiply the ones digits --->2x7=14.
Step 4:-Combine the amount from steps 1,2.and3 writing from left ------>10,914.
Answer 10,914.
Let’s try one more example.
109x105
Step 1:-Begin the answer with 1------------ .>1
Step 2:- Add the ones digits----------- ----->9+5=14.
Step 3:- Multiply the ones digits----------->9x5=45.
Step 4:-Combine the amounts for the steps 1, 2, and 3 .Writing from left to right----->11,445.
Answer is 11,445.
Note :-the sum or the product of ones digits is written in two  ,digits as01,02,03,04,05,06,07,08, and 09.
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SQUARING ANY TWO DIGIT NUMBER ENDING IN ONE OR NINE

Squaring any two digit number ending in 1 or 9 is very interesting and easy method. Two digit numbers ending in 1 are from 11 to 91.Two digit numbers ending in 9 are from 19 to 99.

   The trick is to multiply together the two whole numbers on either side of the number you are squaring, and then add one.
Let’s see how it works.
Problem:-
 31x31 (square of 31)
Step 1:-find the two whole numbers on either of 31 that are 30 and 32
Step 2:-multiply the two whole numbers.30x32=960.
Step 3:- add one to 960 The answer is 961.

Square of 31 is 961.
One more example:
61x61
The two whole numbers on either side of 61 are 60 and 62.
Product of 60 and 62 is 3720.
Adding 1 to 3720 we get 3721.
The answer is 3721.

SQUARE OF ANY TWO DIGIT NUMBER ENDING IN 9..
The method is same as above.
Consider an example of any two digit number ending in 9.

Suppose the number is 19.
Step 1:- find the two whole numbers on either side of 19, that are 18 and 20.
Step 2:- multiply the two whole numbers.
 18x20=360.
Step 3:- adding 1 to 360 ,we get 361.
Square of 19 is 361.

This trick makes the calculation easier by changing one of the numbers into a multiple of 10..

WONDERFUL FOUR

Think of any number.
The number is written in figures and in words.
Suppose we are thinking of 345.
 345 is in figures.
 In words this can be written as ‘three hundred forty five.’
How many letters are in these words?
Count the words.
The letters are 20.
Write 20 in words ‘TWENTY’
Count the letters in twenty.
Number of letters in twenty are 6.
6 is written in words ‘SIX’
Again count the letters in .six.
There are 3 letters in six.
3 is written in words as three.
Three has 5 letters.
5 is written in words as   five.
Five has 4 letters.
4 is written in words a four.
Four has 4 letters.
Try another example.
At last we will come to FOUR.
Does this always happen? What about other languages?

ALWAYS FIVE AND HALF

Think of any digit 1 through 9 in your mind..

Multiply  the number by 11.

Divide the product by its sum of digits.

Answer is always 5.5(five and half)

Example:
Suppose we have chosen the number 8.
8 multiplied by 11.The product is 88.
The sum of digits of 88 is8+8=16.
The product 88 is divided by 16 is 5.5
Answer is 5.5.

Let’s try one more example.
We have chosen the number 7.
7*11=77
Sum of digits =7+7=14.
77/14=5.5.
The answer is five and half.

Triangular Numbers

Pythagoras, the Greek mathematician is known as ‘The father of numbers.’ He explored the mathematical relationship with figurate numbers.
Pythagoras discovered triangular numbers. Triangular numbers are also called Triangle numbers. Pythagoras studied the triangular numbers under figurate numbers in geometry.
First natural number is 1.
Sum of the first two consecutive natural numbers is 1+2 =3
Sum of the first three natural numbers is 1+2+3=6
Sum of the first four natural numbers is 1+2+3+4=10
Sum of the first five natural numbers is 1+2+3+4+5=15
1, 3, 6, 10, 15…are triangular numbers. A    triangle number is the sum of first n natural numbers.
Fredric Gauss discovered the formula for nth triangular number.

Nth triangular number=n (n+1)/2.
For example: 6 th triangular number= 6(6+1)/2=  6(7)/2=  21

Let’s try one more example.
36 th triangular number =36 (36+1) /2
                                         =36 (37) /2= 666.
             666 is called the beast number.

• Every triangular number is a natural number.

• Every triangular number except 1 is divisible by 3 or when divided by 9 the remainder is 1.

• Triangular numbers can never end in 2, 4, 7, or 9.

• Eight times of any triangular number plus 1 is always a perfect square number.

• Nine times of any triangular number plus 1 is always a triangular number.

• Sum of any two consecutive triangular numbers is always a perfect square number.

• Difference between two consecutive squares of any triangular numbers is always a perfect cube of a small triangular number.

• Some triangular numbers are there which reversals are also triangular numbers. The examples are 1,3,6,10,55,66,120,153,171,190 300,595,630,666,820.3003,etc.

Sum Of Ten Successive Fibonacci Numbers

                                     

Sum of any ten successive Fibonacci numbers is eleven times the seventh number.

Observe the following sequence.
1,1,2,3,5,8,13,21,34,55,89,…

In this sequence first two numbers are 1and 1.Third number is the sum of the first two numbers. Fourth number is the sum of the second and the third number. Each subsequent number is the sum of the two numbers before it. This sequence is called Fibonacci number sequence.
 

Fibonacci was an Italian mathematician. He introduced the Hindu-Arabic numeration system
to the west countries. He is well known in the modern world for spreading the Hindu-Arabic
Numeration system. The sequence of numbers is named after him. In Fibonacci sequence the first two numbers are 1 and 1.It is not necessary to take first two numbers be 1and 1.Take any two numbers then form Fibonacci like sequence.

For example
5, 7, 12, 19, 31, 50, 81…

Consider the Fibonacci number sequence of 10 numbers.
1+1+2+3+5+8+13+21+34+55

In this sequence seventh number is 13 and the sum of all ten numbers is 143.143 is the product of 13 and 11.

This rule is applicable for any ten Fibonacci like sequence.

Another example:
5,7,12,19,31,50,81,131,212,343.
In this sequence seventh number is 81.
81x11=891.
Sum of all ten numbers is also 891.

How does this work?
Suppose a and b are any two numbers.
The sequence will be as follows,

a
b
a+b
a+2b
2a+3b
3a+5b
5a+8b
8a+13b
13a+21b
21a+34b

------------
55a+88b

Here seventh number is 5a+8b, when it is multiplied by 11 we get 55a+88b which is the total of all ten numbers.

MAGIC FRACTION OF ODD NUMBER SERIES STARTING FROM ONE

Series of odd numbers starting from 1 is   1+3+5+7+9+11+13+...
Suppose we take the first number of the series as a numerator, we have to take  the Next number in the series as the denominator. Here numerator is 1 and the denominator is 3.  The fraction will be 1/3.

  Now the first two numbers of the series are 1+3 as a numerator.
  Next two numbers are 5+7 .Take this as a denominator.   
  The fraction will be (1+3))/(5+7 )=4 /12.
  Reduced fraction = 1/3.
  In the same way continue the process ( 1+3+5)/(7+9+11)=9/27.
  Reduced fraction = 1/3

  SOME OTHER EXAMPLES:

    (1+3+5+7+9+11)/(13+15+17+19+21+23 )

    =36/108 

Reduced fraction is 1/3.

How many numbers we take from the odd number series starting from 1 as a    numerator, and the next same number of the series as a denominator, the reduced fraction is  always 1/3.