Page 1 of 1

continue here.

http://www.pagalguy.com/forums/quantitative-ability-and-di/number-system-questions-and-discussions-t-43728

http://www.pagalguy.com/forums/quantitative-ability-and-di/mind-blowing-concepts-at-quant-t-10313

Commenting on this post has been disabled by the moderator.

First express the given number 'N' in the form of f1^a x f2^b x f3^c.. where f1,f2,f3...fn are the prime factors of the given number 'N and a,b,c..are positive integers.

Then the

**Total no. of factors of 'N' is (a+1) x (b+1) x (c+1)**....The sum of all factors (

**including 1 and the 'N' itself**) is given by**[(f1^(a+1))/(f1-1)] x [(f2^(b+1))/(f2-1)] x**

**[(f3^(c+1))/(f3-1)]**....

The different ways of expressing the 'N with two different factors can be done in

**[(a+1) x (b+1) x (c+1)**

**... -1 ] / 2 ways.**

The different ways of expressing the 'N with two factors (including N^

*1/2*x N^*1/2*) can be done in**[(a+1) x (b+1) x (c+1)**

**... +1 ] / 2 ways.**

The different ways of expressing the 'N with two co-primes can be done in

**2^(A-1) ways.**

Where A is the number of distinct prime factors of 'N'.

The number of co-prime of N that are less than N is given by :

**N(1-(1/f1)) x (1-(1/f2))...**

The sum of co-primes of N less than N is given by :

**(N^**

*2*/2) x (1-(1/f1)) x (1-(1/f2))... which can be simplified as (N/2)**x ( No. of co-primes of N less than N)**

------------------------------------------------------------------------------------------------

Let us see with an example :

Let N be

**48**48 can be expressed as 2^

*4*x 3^*1*Here f1=2 and f2=3 and also a=4 and b=1.

So, No of factors of 48 is (4+1) x (1+1) = 5 x 2 =

**10**.Sum of all factors of 48 is ((2^(

*4+1*) -1)/2-1) x ((3^(*1+1*)-1)/3-1)= ((2^*5*-1)/1) x ((3^*2*-1)/2) 1 = ((32-1)/1) x ((9-1)/2) = 31 x 4 =**124**.The different ways of expressing the 48 with two different factors can be done in

[ (4+1) x (1+1) -1 ]/2 = 9/2 =

**5**ways ( Just add 1 if the numerator comes as an odd value and then divide by 2).The different ways of expressing the 48 with two factors can be done in

[ (4+1) x (1+1) + 1 ]/2 = 11/2 =

**6**ways ( Just add 1 if the numerator comes as an odd value and then divide by 2).The different ways of expressing the 48 with two co-primes can be done in

For 48, there are two distinct prime factors and therefore A is 2 here for us. So

2^(2-1)=

**2**waysThe number of co-prime of 48 that are less than N is

48 x ( 1-(1/2)) x (1-(1/3)) = 48 X 1/2 X 2/3 =

**16**

The sum of all co-primes less than 48 is :

(48/2) x 16

**= 384**

**Read '^' as 'to the power of' and 'x' as 'multiplied with' !!**

Hope I have made you to understood the relationship between numbers and its factors in a detailed manner and do revert me back for any clarifications/doubts/corrections.

All the best puys !!

Commenting on this post has been disabled by the moderator.