Question:
How many positive integral solutions are possible for the equation X2– Y2 = 160?

Answer:
To find the number of positive integral solutions for the equation X2 – Y2 = 160, we need to analyze the given equation and express it in a suitable form.

Analysis of the Equation:
The given equation is a difference of squares, which can be expressed as (X + Y)(X - Y) = 160.

Prime Factorization:
To find the positive integral solutions, we can factorize 160 into its prime factors.

160 = 2 × 2 × 2 × 2 × 5 = 2^4 × 5

Factors of 160:
Now, let's list down all the factors of 160.

1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160

Arranging Factors into Pairs:
Since (X + Y)(X - Y) = 160, the factors of 160 can be arranged into pairs in order to satisfy the equation.

Case 1: X + Y = Factor and X - Y = Factor
- For example, if X + Y = 2 and X - Y = 80, we can solve these equations simultaneously to find X and Y.
- Solving these equations, we get X = 41 and Y = -39. But as per the given condition, we need to consider only positive integral solutions. Therefore, this solution is not valid.

Case 2: X + Y = Factor and X - Y = Another Factor
- For example, if X + Y = 4 and X - Y = 40, we can solve these equations simultaneously to find X and Y.
- Solving these equations, we get X = 22 and Y = 18. This is a valid positive integral solution.

Case 3: X + Y = Another Factor and X - Y = Factor
- For example, if X + Y = 80 and X - Y = 2, we can solve these equations simultaneously to find X and Y.
- Solving these equations, we get X = 41 and Y = 39. This is a valid positive integral solution.

Summary:
From the above analysis, we can conclude that there are 2 positive integral solutions for the equation X2 – Y2 = 160, which are:
1. X = 22, Y = 18
2. X = 41, Y = 39

Therefore, there are a total of 2 positive integral solutions for the given equation.