Pythagoras triplet

Pythagoras-

Born-570 BCE, Samos(Greece).
Died-500-490 BCE, Metapontum(Italy).
He was a Greek philosopher, mathematician. Pythagoras is most famous for his concept of geometry.

Pythagoras

          a²+b²=c²

        pythagoras theorem


  





Pythagoras triplet theorem:-Three natural numbers a, b and c are such that  a²+b²=c², Then a, b and c are called Pythagoras triplet.

                                      Ex-(3,4,5)
            
                                      3²+4²=5²
                                      Ex- (5,12,13)

                                      5²+12²=13²

General form-     2m, m²-1, and m²+1

Properties of odd numbers or even numbers

1.The product of two consecutive even numbers-

                    8×10=9²-1

         

                    12×14=13²-1

    

                    14×16=15²-1

                    16×18=17²-1

      

                    ......................

                   106×108=107²-1

      note-    (middle term)²-1

The great mathematician and astronomer.

Aryabhata was an extraordinary teacher and scholar.

He was a famous Indian mathematician and astronomer, born in a place called Taregna, in Bihar which literally means songs of stars in Bihari, is a small place situated near Patna.

                   Aryabhata(476-550A.D.)

Works of Aryabhata-

1.value of pi.  2. He discussed the idea of sin.

3. Describes the solar and lunar eclipses scientifically 4.invention of zero.

Properties of '0'

Properties of zero(0):-

1. If any number is multiplied by zero then becomes zero.
Ex- 0×5=0
       0×8=0
       0×45=0
2.If zero is added any numbers then numbers remain unchanged.
Ex- 0+2=2
       0+6=6
       0+50=50
3.If any number is divided by Zero then it becomes undefined.
            Ex-5/0,14/0(undefined).
4.If zero is divided by any numbers then it becomes zero.Ex-0/9=0,0/45=0

Properties of even numbers and odd numbers

properties 1. Odd numbers

→ 1+3=2²=4               (number of terms 2)

     1+3+5=3²=9           (number of terms 3)

     1+3+5+.........=n²     (number of terms n)
 
  properties 2.Even numbers

    →2+4=2(2+1)=6            (number of terms 2)

      2+4+6=3(3+1)=12       (number of terms 3)

    2+4+6+...........=n(n+1)  (number of terms n)


Properties 3.

          →   1+2+1=2²=4           (middle term 2)

               1+2+3+2+1=3²=9     (middle term 3)

          1+2+3+4+3+2+1=4²=16 (middle term 4)
 1+2+3.....+n+..........+3+2+1=n² (middle term n)

properties 4.

→         1²=1    (one digit)

            11²=121  (two digits)

         111²=12321 (three digits)

     1111²=1234321 (four digits)

Even numbers and odd numbers

EVEN NUMBER:-An integer which is completely divisible by '2'(unit place should be 0,2,4,6,8) is called an even number.Ex-0,4,6,8,10,1982,-16,-1470............
ODD NUMBER:-An integer which is not completely divisible by '2'(unit place should be 1,3,5,7,9) is called an odd number.Ex- 1,3,5,7,9,11,-25,-123,1427........

Important properties-

1. Difference between any two consecutive even or odd numbers is 2.
2. Sum of even numbers is an even number.ex- 4+8=12, 24+12+8=44.
3. Sum of any two odd numbers is an even number.ex- 1+3=4,7+9=16,11+(-3)=8.
4. Sum of an even number and an odd number is an odd number.ex-1+2=3,7+22=29,15+(-8)=7.

Properties of numbers

1.which of following is a perfect number.

A.2                  B.28

C.12                D.19

2.How many prime numbers are exist between 50 and 100.

A.5                 B.8

C.10               D.15

3.Which of following is an even prime number.

A.2                B.4

C.5                D.3

4.Which of following is a composite number.

A.2              B.3

C.4              D.5

ans:-B,C,A,C.

Properties of numbers

Composite number:-A natural number which has more than two factors is called a composite number Ex-4-factors(1,2,4),6-factors(1,2,3,6).
Note:-
1.First composite number is 4.
2.1 is neither composite nor prime number.

Properties of number

Prime numbers:-A natural number which has exactly two factors(1 and itself) is called a prime number Ex:-2-factors(1,2),
3-factors(1,3),19-factors(1,19).
Note:-1.There are 15 prime numbers between 1 and 50.
2.There are 10 prime numbers between 50 and 100.
3.There are 25 prime numbers between 1 and 100.
4.There are 46 prime numbers between 1 and 200.
5.First prime number is 2.
6.Only 2 is an even prime number.

Properties of number

Twin prime numbers-A pair of prime numbers is such that they are differ by 2(difference is 2) is called twin prime numbers.Ex-(3,5),(11,13),(29,31),(41,43),

etc.

1.Which of followings are twin prime numbers.

A.(43,45)         B.(71,73)

C.(7,9)              D.(2,3)

Properties of numbers

Perfect numbers-A natural number whose sum of factors is equal to two times of itself is called a perfect number.ex-factors of 6 is 1,2,3,6. Sum of factors=1+2+3+6=12.

6×2=12,so 6 is a perfect number.

Factors of 28 is 1,2,4,7,14,28.

Sum of factors=1+2+4+7+14+28=56.

28×2=56,so 28 is a perfect number.


Co-prime numbers:-A pair of two numbers a and b are such that HCF(a,b)=1,then a and b are called co-prime numbers.
Ex-(2,3),(5,7),(7,9),(11,17),(10,17).

Properties of Co-prime numbers-

1.          LCM(a,b)=a×b.

  Ex- LCM(4,5)=4×5=20.

         LCM(11,15)=11×15=165.

          LCM(21,29)=21×29=609.

2.          HCF(a,b)=1


      Ex-       HCF(12,17)=1

3.Divisibility-If we take a number 15.

      Here,15=3×5, HCF(3,5)=1,so we say that         3 and 5 are Co-prime numbers to each-        other.
Divisubility of 15- A number which is divisible by 3 and 5 both ,then they will be divisible by 15.

4.Solving question of LCM and HCF-

(1)LCM of two numbers is 24 and HCF is 2,then how many pairs of number are possible?
  Sol-numbers-2x,2y

   LCM=2xy

      a/q  2xy=24

               xy=12=1×12=3×4=2×6

          Here,(2,6) is not a Co-Prime pairs,so                number of pairs is 2.