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.