Number Of Solutions

This is a typical pattern. If you can remember different formats of this specific problem, cracking questions is just a matter of few seconds. Let's have a look.

#Problem 13.1

Let n ≥ 1 be an integer and let t denote the number of positive integer divisors of n^2. Show
that the number of positive integer solutions (a, b) of the equation 1/a−1/b = 1/n is precisely
equal to (t − 1)/2.

Solution scheme and approach:
1/a-1/b=1/n..............(1)
From the above equation it's clear that 1/a>1/n =>n>a
Now we can consider an integer x such that
a=n-x
From equation number (1) we can also write
1/b=1/(n-x)-1/n=x/(n-x)*n
=>b=(n^2-nx)/x=n^2/x-n
To count the total number of solutions we have to count total numbers of possible values of x.
As b is an integer x has to be an integer which must be a factor of n^2 such that x
Now think, n^2 is a perfect square which has a total of t divisors. out of these factors one is n another (t-1)/2 is less than n each of them is paired with other (t-1)/2 number of factors which are greater than n.
[Suppose for 16=4x4=2x8=1x16]
Hence, Total number of solutions are (t-1)/2. Q.E.D.
[N.B: Total number of ordered pair(a,b) would be (t-1)]

Food for thought: 
Let n ≥ 1 be an integer and let t denote the number of positive integer divisors of n^2. Show that the number of positive integer solutions (a, b) of the equation 1/a+1/b = 1/n is precisely equal to t /2.

#Problem 13.2

Find the smallest positive integer n with the property that the equation 1/a − 1/b = 1/n has exactly 2010 different solutions in positive integers a and b.

(a)6^2010,(b)2^2010,(c)1,(d)10^2010,(e)Cannot be determined  

Solution scheme and approach

From the above equation it's clear that 

n^2 should have t number of divisors where (t-1)/2=2010=>t=4021

Suppose 

n=p1^a*p2^b*p3^c*...

=>n^2=p1^2a*p2^2b*p3^2c*...pn^2n

It has (2a+1)*(2b+1)*(2c+1)*....*(2n+1)=4021 number of divisors 

Now the catch is 4021 is a prime number.=>n^2=p1^2a

=>(2a+1)=4021

=>a=2010

As we need to find out smallest possible value of p1 that means p1=2

So the required number is 2^2010

Hence, option (b) is the right option. 

#Problem 13.3

If Ramu and Krishna work on alternate days to complete a work, then the work gets completed in exactly 24 days. If R and K denote the number of days required by Ramu and Krishna respectively to complete the work independently, then how many ordered pairs of integral values of R and K are possible? 
(a)14, (b)8, (c)15, (d)7,(e)16                                                [Career Launcher Mock CAT]
Solution scheme and approach
Ramu and Krishna work 1/R and 1/K th work per day respectively
According to the question as they are working on alternative day.We may say if they work together work must be completed on 12th day
=>12(1/R+1/K)=1
=>(1/R+1.K)=1/12
As, 12^2=2^4*3^2 has [(4+1)*(2+1)]/2=15/2 number of solutions.So total number of ordered pair(R,K) solutions are 15. (refer Food for thought) 
Hence, Option (c). 

Hope, this session has helped you a little bit to clear these things.
All the best!!