Geometry

Basics of Geometry

1.Line Segment: A line segment consists of two endpoints. we can measure the length of the segments.

2.Line: A line doesn't have any endpoints.

3.Intersection: Lines passing through a single point.

4.Ray: A ray is a portion of a line. It starts at one point (called starting point)
and goes endlessly in a direction.

5.Sides,vertices and Diagonals: 

6.Quadrilaterals:A four sided polygon is a quadrilateral.It has 4 sides and 4 angles.

   

Divisibility

1.Divisibility by 10: If a number has 0 in the one's place then it is divisible by 10.

2.Divisibility by 5: A number which has either 0 or 5 in it's one's place is divisible by 5.

3.Divisibility by 2: A number is divisible by 2 if it has 0,2,4,6 or 8 in it's one's place.

4.Divisibility by 3: If sum of the digits is a multiple of 3, then the number is divisible by 3
                               ex: 112461 = (1+1+2+4+6+1)=15 is divisible by 3 , so 112461 is divisible by 3.

5.Divisibility by 6: If a number is divisible by both 2 & 3 then it is divisible by 6.

6.Divisibility by 4: A number with 3 or more digits is divisible by 4 if the number formed by its last two digits (i.e; ones & tens) is divisible by 4.
 Divisibility for 1 0r 2 digit number by 4 has to be checked by actual division.

            ex: 12458936 = last 2 digits =36 which is divisible by 4 , so 12458936 is divisible by 4

7.Divisibility by 8: A number with 4 or more digits is divisible by 8 if the number formed by the last     3 digits is divisible by 8.
   The divisibility for numbers with 1,2 or 3 digits by 8 has to b checked by actual division.

8.Divisibility by 9: If the sum of the digits is divisible by 9, then the number itself is divisible by 9.

9.Divisibility by 11: Find the difference between the sum of the digits at the odd place (from the right) and the sum of digits at even place (from the right) of a number.
If the difference is either 0 or divisible by 11, then the number is divisible by 11.
    ex: 61809
         => (9+8+6)-(0+1) = 23-1=22 .
          Since 22 is divisible by 11, so 61809 is divisible by 11.

10.Divisibility by 7:   
 For 3 digits numbers: Subtract twice the last digit from the number formed by the remaining digits. If the remaining number is multiple of 7 then it is divisible by 7.
ex: 651
  =>65-(1*2)
  =>63
Since 63 is divisible by 7, so 651 is divisible by 7.
For 4 digits and above numbers: Divide the number into two groups of 3 digits (starting from right) and find the difference between the sum of the numbers in odd and even places. If the difference is 0 or divisible by 7, it is divisible by 7.
 ex:4537792
  =>(792+4)-(537)= 259, which is divisible by 7 .

11.Divisibility by 13:
     For 3 digits numbers: Multiple last digit with 4 and add with the remaining number, the number formed is divisible 13 then the entire number is divisible by 13.
   ex: 273
     =>27+(3*4)
     =>27+12=39 , which is divisible by 13.
      For 4 digits and above numbers: Divide the number into two groups of 3 digits (starting from right) and find the difference between the sum of the numbers in odd and even places. If the difference is 0 or divisible by 13, it is divisible by 13.
 ex:579488
  =>(579)-(488)= 91, which is divisible by 13 .









   

Fundamentals

1.Predecessor & Successor:
 The Predecessor of 16 = (16-1)= 15.
 The Successor of 16 = (16+1)= 17.
Natural Numbers: 1,2,3,4...
Whole Numbers: 0,1,2,3,4...

Note: Zero is called the identity of addition or additive identity for whole no.
          ex: 7+0=7
               15+0=15
               10+0=10
          One is the identity for multiplication of whole no. or multiplicative identity for whole no.
          ex: 7*1=7
                15*1=15
               
 2.Factors:
    a. Factors of a number is an exact divisor of that number.
    b.Factors of a number are always less than or equal to that number whose factor is said to be found.
     ex:
      Factor of 12: 1,2,3,4,6,12.

    Perfect Number: A number for which the sum of all its factors is equal to twice the number is called a perfect number.
    ex:
          Factors of 6 = 1,2,3,6.
          Sum of factors of 6 = 1+2+3+6=12.
          Therefore 6 is the Perfect number.

Note: Important Concept while finding factors for any number, while finding the factor starts with 1,2,3.. and stops where the number gets repeating.
  ex:
    Factors of 36: (1*36),(2*18),(3*12),(4*9),(6*6) Now stop dont go further
    The factors are : 1,2,3,4,6,9,12,18,36.

3.Prime and Composite Number:
   A prime number is the number whose factors are only 1 and itself are called prime numbers.
   ex: 2=1,2
         3=1,3
         4 =1,2,4
 So, 2 and 3 are prime numbers.
    Composite numbers are those numbers having more than two factors.
    ex: 10=1,2,5,10
     
     Even No.: 2,4,6,8.
     Odd No.: 1,3,5,7.

Interesting Facts:

• 2 is the smallest Even Number and smallest prime number.
• Every prime number except 2 is odd.