Number System-Integral Solutions: Shortcuts and Tricks

 This concept is important for CAT and XAT.

Questions based on Integral solutions are frequently asked in CAT, XAT and other MBA entrance exams. Apart from Number system, this kind of questions may belong to areas like Algebra or Permutation & Combinations. Such questions feature in the forms discussed below:

Question 1: How many positive integral solutions are possible for the equation X²- Y²= N?

Many of us tend to leave this question or try solving this equation using algebraic formula. In this article, we have discussed the basic trick to solve this type of question.

When we are asked to calculate how many positive integral solutions are possible for the equation X²- Y²= N, there can be 4 cases.

Case 1: N is an odd number and not a perfect square

Case 2: N is an odd number and a perfect square

Case 3: N is an even number and not a perfect square

Case 4: N is an even number and a perfect square

Let us discuss these cases in detail.

Case 1: N is an odd number and not a perfect square.

In this case, total number of positive integral solutions will be= (Total number of factors of N) / 2

Example: How many positive integral solutions are possible for the equation X²- Y²= 105?

Solution: Total number of factors of 105 is 8.

So, total number of positive integral solutions = 8/2 = 4

Case 2: N is an odd number and a perfect square.

In this case, total number of positive integral solutions will be = [(Total number of factors of N) - 1] / 2

Example: How many positive integral solutions are possible for the equation X²- Y²= 225?

Solution: Total number of factors of 225 is 9.

So, total number of positive integral solutions = (9-1)/2 = 4

Case 3: N here is an even number and not a perfect square

In this case, total number of positive integral solutions will be = [Total number of factors of (N/4)] / 2

Example: How many positive integral solutions are possible for the equation X²- Y²= 120?

Solution: Total number of factors of 30 is 8 ( as N=120 and N/4=30)

So, total number of positive integral solutions = 8/2 = 4

Case 4: N here is an even number and a perfect square

In this case, total number of positive integral solutions will be ={[Total number of factors of (N/4)] - 1 } / 2

Example: How many positive integral solutions are possible for the equation X²- Y²= 400?

Solution: Total number of factors of 100 is 9.

So (9-1)/2 = 4 positive integral solutions

Another frequently asked question is:

Question 2: How many positive integral solutions are possible for the equation 1/X + 1/Y= 1/N (N here is a natural number)?

In this case, total number of positive integral solutions will be = Total number of factors of N²

Example: How many positive integral solutions are possible for the equation 1/X + 1/Y= 12?

Solution: N=12 =>N²=144.

Total number of factors of 144 is 15.

So, total number of positive integral solutions=15.