p : if two integers a and b are such that `a gt b` then a - b is always a positive integer.
q : If two integers a and b are such that a - b is always a positive integer, then `a gt b `
Which of the following is true regarding statements p and q?

Options

A. (a) ~ p is converse of q
B. (b) ~ p is contrapositive of p
C. (c) p is converse of q
D. (d) p is contrapositive of q

Answer

To solve the problem, we need to analyze the two statements \( p \) and \( q \) and determine their logical relationships.

### Step 1: Understand Statement \( p \)
Statement \( p \) says:
- If two integers \( a \) and \( b \) are such that \( a > b \), then \( a - b \) is always a positive integer.

This can be broken down into:
- Hypothesis: \( a > b \)
- Conclusion: \( a - b \) is a positive integer.

### Step 2: Understand Statement \( q \)
Statement \( q \) says:
- If two integers \( a \) and \( b \) are such that \( a - b \) is always a positive integer, then \( a > b \).

This can be broken down into:
- Hypothesis: \( a - b \) is a positive integer.
- Conclusion: \( a > b \).

### Step 3: Identify the Contrapositive of Statement \( p \)
The contrapositive of a statement is formed by negating both the hypothesis and conclusion and swapping them.

For statement \( p \):
- Hypothesis: \( a > b \) becomes \( a \leq b \) (negation).
- Conclusion: \( a - b \) is a positive integer becomes \( a - b \) is not a positive integer (negation).

Thus, the contrapositive of \( p \) is:
- If \( a - b \) is not a positive integer, then \( a \leq b \).

### Step 4: Identify the Contrapositive of Statement \( q \)
For statement \( q \):
- Hypothesis: \( a - b \) is a positive integer becomes \( a - b \) is not a positive integer (negation).
- Conclusion: \( a > b \) becomes \( a \leq b \) (negation).

Thus, the contrapositive of \( q \) is:
- If \( a \leq b \), then \( a - b \) is not a positive integer.

### Step 5: Identify the Converse of Statement \( p \)
The converse of a statement is formed by swapping the hypothesis and conclusion.

For statement \( p \):
- The converse is:
- If \( a - b \) is a positive integer, then \( a > b \).

This matches statement \( q \).

### Step 6: Identify the Converse of Statement \( q \)
For statement \( q \):
- The converse is:
- If \( a > b \), then \( a - b \) is a positive integer.

This matches statement \( p \).

### Conclusion
From the analysis:
- Statement \( q \) is the converse of statement \( p \).
- Therefore, the correct answer is option \( c \): \( p \) is the converse of \( q \).