How To Solve Problems On Factors

Problems based on factors are a number of times asked in CAT exam and various other management exams.

So this is important to learn this topic right from the basics.

If  I say When 2 No. are multiplied then the resultant value is called their Product.

eg.    when we multiply 3 and 4 we get 12 as their Product

3x4=12

In the same Manner when we multiply 2,7 and 5 we get 70 as their Product.

So From the Above examples we can get a clear definition of what Factor is?

Factor is a number which Divides a given number exactly.

As in the above case 3 and 4 are the factors of 12.

But it is also be noted that 1 and the number itself is always the factor of the given number. That is in case of 12 it as Factors 1,2,3,4,6 and 12. But in Exam nobody is going to Ask you What Factor is? he will ask you Problem related to Factors.

FINDING NO. OF FACTORS:

In this we need to find No. of factors of a given Number.

Like find No. of Factors of 24. It can Be counted easily

Factors of 24 are 1,2,3,4,6,8,12,24 in all total no. of factors of 24 are 8.

As this was an easy question so we can count them but what happens when we need to find the factors for a bigger no. the job becomes tedious. So in order to make our job easy in mathematics we are having a formula for this type of problem.

STEPS

1. First Step is to do Prime Factorisation of that Number. let us take the example suppose the Number be 'A' we know that on prime factorization of the number we get as A=B^pxC^q and so on.


2. In the second step we just need to multiply the power of the B,C by adding 1 to them            i.e:   (p+1)x(q+1) this our answer.

Now let us solve it with an example take 24, we know the factorization of 24 is 2³x3¹   this means that 2 is having a power of 3 whereas 3 is having power 1. So 3 and 1 are our p and q so

SUM OF FACTORS=  (3+1)x(1+1)= 4x2=   8

In this way we can solve the problems Based on sum of factors.

FINDING SUM OF FACTORS:

In above we learn about how to find the number of factors of a given number. In this part we will learn how to find the sum of factors of the number. For Finding the sum of factors we will follow the step 1 as we did in above i.e prime factorization

suppose the number be 'A' we find the prime factorization of A = B^pxC^q           In this case to Find the sum of factors of the Number we Have a formula

In this formula we add +1 to the power of the factor  i.e B^(p+1)
and then subtract 1 from the Number i.e  (B^(p+1)-1)(C^(q+1)-1)..........and so on

and then divide the whole by the products of the factors B,C... subtracting 1 from Them.

So the Final Formula comes out to be SUM OF FACTORS=  (B^(p+1)-1)(C^(q+1)-1)....../(B-1)(C-1).......

SOLVING A PRACTICAL PROBLEM

Q. Find the sum of the factors of 32?

A. If we go by putting the values in the formula

Factorization of the number 32= 2⁵ i.e 2 with a power of 5

SUM OF FACTORS= (2⁽⁵⁺¹⁾-1)/(2-1)               So it comes out to be 2^6 i.e 2 with a power of 6 which is equal to 64.

So the Answer for the above Problem is 63.

FINDING THE PRODUCT OF THE FACTORS:

We have done the concepts on finding the factors, sum of the factor. Now we will discuss how to find the product of the factors of a number.

for this type of problem we are having a very easy formula i.e PRODUCT OF FACTOR= (A)^(n/2)

Where,

A = Given Number.

n = Number of Factors.

Know Let us solve a problem based on this.

Q. Find the product of the Factors of 24?

A. first we will find the number of factors of 24= 2³x3¹

Number of factors =(3+1)x(1+1) = 8.

PRODUCT OF THE FACTORS =  (24)^(8/2)
So  24^4 is our Answer.

FINDING NO. OF ODD FACTORS:

In order to find no. of factors of a number we will do prime factorization of the number.
let us take example 24 be our number,

Prime Factorisation of 120= 2³x3¹x5¹

we know 2 is a even number. In order to find Odd no. of Factors we will the number 2 and calculate the number of factors other than that.

NO. OF FACTORS = (power of 3 + 1)x(power 0f 5 + 1)

i.e (1+1)x(1+1)= 4  So there are 4 odd factors in this number.

If you Want to find number of Even factors you can do it by subtracting Odd No. of factors from the total No. of Factors

Even No. of factors = Total No. of factors - Odd No. of factors.

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