This is my first attempt in writing a note. Please give me your valuable comments and suggestion; of course correct me wherever I am wrong. This is my own writing and all the examples are original one.

"Factors" are the numbers you multiply to get another number.

Example: The factors of 20 are

20 and 1, because 20 ×1 = 20 .

10 and 2, because 10 ×2 = 20 .

The prime factorization of a number includes ONLY the prime factors, not any products of those factors. For 20 we ‘ll write it as 2² x 5

2 and 5 are called the proper factors of 20 while others factors like (1, 10), (2, 10). . .etc are called improper factors.

1. Number of ways in resolving a composite numbers into two factors

(In Kundan/Pandey a formula is given for this w/o any explanation and I don’t think that right approach to solve these problems, its too difficult to remember all those short cuts and its confusing also)

Different factors of 144 are 12 x 12 ; 48 x 3; 36 x 4. . .etc

The question is to ask how many such pair of two factors is possible

Do the prime factorization and write the number only in prime factors

(for N = a^m x bⁿ x c^p x . . .      Number of divisors of N is (m+1) (n+1) (p+1) page @ http://www.cat4mba.com/node/6030)

Number of divisors of 144 are (2+1)(4+1) = 15

i.e 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144

144 is a square number and its square root is 12. So one pair is 12 x 12.

Now we are left with 14 numbers, so possible number of pairs = 7

And total number of ways = 7 + 1 = 8

(1 x 144, 2 x 72, 4 x 36, 8 x 18, 16 x 9, 48 x 3, 24 x 6, 12 x 12 )

Lets consider another number, which is not square – 40

So the total number of ways of resolving it into two factors = 8/2 = 4

If the number of divisors of a composite number is = even (say X)

If the number of divisors of a composite number is = odd (say X)

Formula given in Books (kundan/pandey)

If N = A₁^m1 x A₂^m2 x A₃^m3

Number of ways = 1/2 (m1 +1) (m2 + 1) (m3 + 1)

2. Number of ways in resolving a composite numbers into two factors that are prime to each other

Let the number N has n different prime numbers then the number of ways in which N can be expressed as the product of two positive integers which are prime to each other are

How??? Try to think

Example: The number 330 (= 2 x 3 x 5 x 11) can be expressed as a product of two factors in 8 (2³) different ways, which are prime to each other.