**Basics of Factors:**

What are factors? What does the term mean? If we go by the Latin translation of the word, it means ‘who/which acts’. Well, in our mathematical sense, factors are the ones that act for sure, but in this case they act on numbers.

** Mathematically speaking:** If any integer, say P, is divisible by another integer say Q an exact number of times then P is said to be a multiple of Q and Q is the factor of P.

**Number of Factors of a given number:**

Number of factors can be expressed by following steps:

- First write down the number in prime factorisation form i.e. a
^{p}b^{q}c^{r }(where a,b,c, are prime numbers and the p,q,r are natural numbers as their respective powers) - Number of factors can be expressed as (p+1)(q+1)(r+1).
- Here factors include 1 and the number itself.

**The points above explained through a example:**

Let us take an example of a number N = 38491200=2^{6} 3^{7} 5^{2}

Now observe some facts about the number of factors (we will solve the problem step by step):

**Step 1:** Prime factorisation, so N=38491200=2^{6} 3^{7} 5^{2} 11

Power of 2 as 2^{0 }, 2^{1}, 2^{2 } ,2^{3},2^{4},2^{5},2^{6}( 6+1=7)ways,

Power of 3 as 3^{0 }, 3^{1}, 3^{2 },3^{3},3^{4,},3^{5},3^{6},3^{7}( 7+1=8)

Power of 5 as 5^{0 }, 5^{1}, 5^{2 }( 2+1=3)ways

Power of 11 as 11^{0 }, 11^{1} ^{ }( 1+1=2)ways

**Step 2:**Hence, the number of factors is given by (6+1)(7+1)(2+1)(1+1)=7x8x3x2=336

**Example 1**:Find the **number of factors** of 2^{4}3^{1}5^{2}7^{2}.

**Solution:** As we can see the above number has 2,3,5,7 which all are prime numbers and they have 4,1,2,2 as their powers so the number of factors of the given number are (4+1)(1+1)(2+1)(2+1)= 90

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**Example 2**:Find the **number of factors** of 1440 ?

**Solution:** We first factorize 1440.

1440 = 2^{5}3^{2}5^{1}

2,3,5 are prime numbers and they have 5,2,1 as their powers so the number of factors of the given number are (5+1)(2+1)(1+1)= 36