What is the value of a two-digit number in which digit at tens place is greater than digit at unit place?
A. The sum of the digits is 5.
II. The difference of the squares of the digits is 15.
C. The difference of their digits is 3.

Options

A. A and B together are sufficient
B. B and C together are sufficient
C. C and A together are sufficient
D. Any one pair of A and B, B and C and A is sufficient

Explanation

To solve the problem, we need to determine the value of a two-digit number where the digit in the tens place is greater than the digit in the units place. We will analyze the given statements one by one to see if they provide sufficient information to find the digits.

Let's denote the two-digit number as \(xy\), where \(x\) is the digit in the tens place and \(y\) is the digit in the units place. We know from the problem that \(x > y\).

### Step 1: Analyze Statement I
**Statement I:** The sum of the digits is 5.
This can be expressed mathematically as:
\[ x + y = 5 \]

### Step 2: Analyze Statement III
**Statement III:** The difference of their digits is 3.
This can be expressed mathematically as:
\[ x - y = 3 \]

### Step 3: Solve the Equations from Statements I and III
Now we have the two equations:
1. \( x + y = 5 \)
2. \( x - y = 3 \)

We can solve these equations simultaneously.

Adding the two equations:
\[ (x + y) + (x - y) = 5 + 3 \]
\[ 2x = 8 \implies x = 4 \]

Now substituting \(x = 4\) back into one of the equations, let's use \(x + y = 5\):
\[ 4 + y = 5 \implies y = 1 \]

### Conclusion from Statements I and III
From Statements I and III, we find:
- \(x = 4\)
- \(y = 1\)

Thus, the two-digit number is \(41\).

### Step 4: Analyze Statement II
**Statement II:** The difference of the squares of the digits is 15.
This can be expressed mathematically as:
\[ x^2 - y^2 = 15 \]

Using the identity \(x^2 - y^2 = (x - y)(x + y)\), we can substitute:
- \(x - y = 3\) (from Statement III)
- \(x + y = 5\) (from Statement I)

Thus:
\[ (x - y)(x + y) = 3 \cdot 5 = 15 \]

This confirms that Statement II is also satisfied by the values of \(x\) and \(y\) we found earlier.

### Final Conclusion
All three statements together are consistent and lead to the same digits \(x = 4\) and \(y = 1\). Therefore, the value of the two-digit number is \(41\).