**For doing prime factorization of any number, we should know about the following important points:**

**1)**. The numbers 1 and 0 are neither composite nor prime numbers and hence, they can never be a part of prime factors of any given number.

**2)**. Each and every number will always be factor of 0.

**3)**. The only number 1 will always has one and only one factor, which is one itself.

**4)**. To do prime factorization, we will keep on factoring the given number, until we figure out all the prime factors involved in it. If we have got a composite number, then we should remember that it can again be factorized further to get a prime factor again. Thus, the procedure of prime factorization only ends, whenever; we have found all its factors as prime numbers or factors.

**5)**. The set of prime factors of a given number is always unique, that is the same set of prime numbers can not be used to factorize some other number. This statement is also known as the famous

Fundamental Theorem of Arithmetic. As the multiplication of number is commutative, thus it does not matter in which order the factors are being multiplied.

**Now we will list some of the important formulas of the prime factorization method which are listed below:**

Formula 1: The number of prime factors of a number is always found by adding one to the exponents of the prime factors, which are written in the exponent notation and then multiplying the adding number with each other.

__ For example:__ 72 can be prime factorized by giving us the prime factors as

72 = 2

^{3} x 3

^{2}Now, the value of exponent of 2 is 3, now adding 1 to 3 is 4.

Adding 1 to exponent of 3, which is 2 gives us 3.

Now multiplying 4 x 3 = 12. Hence, the total number of factors that number 72 has is 12, which is true as 1, 2, 3, 4, 6, 8, 9, 12, 16, 24, 36, and 72, are all the factors of 72, which if counted are 12 in numbers.

**Formula 2:**

Euler’s Formula:

This formula is based on the famous Euler’s Algorithm which starts from the following steps:__Step 1:__ Let n be an odd number whose prime factorization has to be done then, if

n = a

^{2} + b

^{2}Also assume that n can also be expressed as a sum of the squares of two other numbers such that,

n = c

^{2} + d

^{2}__Step 2:__ Now, a

^{2} + b

^{2} = c

^{2} + d

^{2}, this implies, a

^{2} – c

^{2} = d

^{2} – b

^{2} or

(a + c) (a - b) = (b + d) (b - d).

__Step 3:__ If the

Greatest common factor of a – c and d – b is k, then we will write,

a – c = k * l, d – b = k * m, and G.C.D of (l, m) is equal to1.

__Step 4:__ Now, l and m are

relatively prime numbers to each other, and hence, l (a + c) = m (d + b), and therefore, a + c is can be divided by m and hence,

a + c = m * n and d + b = l * n.

__Step 5:__ Then n would be equal to

$[(\frac{k}{2})^{2}+(\frac{n}{2})^{2}]$$(m^{2}+l^{2})$.