Write the converse and contrapositive of the following statement:
A positive integer is prime only if it has no divisors other than 1 and itself.

Answer

To solve the problem of writing the converse and contrapositive of the statement "A positive integer is prime only if it has no divisors other than 1 and itself," we can follow these steps:

### Step 1: Rewrite the Original Statement
The original statement can be rewritten in an "if-then" format.
- Let \( p \) be the statement "A positive integer is prime."
- Let \( q \) be the statement "It has no divisors other than 1 and itself."

Thus, we can express the original statement as:
"If \( p \) (a positive integer is prime), then \( q \) (it has no divisors other than 1 and itself)."

### Step 2: Write the Converse
The converse of a statement switches the hypothesis and the conclusion. Therefore, the converse of the original statement is:
"If \( q \) (it has no divisors other than 1 and itself), then \( p \) (a positive integer is prime)."

### Step 3: Write the Contrapositive
The contrapositive of a statement negates both the hypothesis and the conclusion, and then switches them. Thus, the contrapositive of the original statement is:
"If not \( q \) (it has divisors other than 1 and itself), then not \( p \) (it is not prime)."

### Final Statements
1. **Converse**: If a positive integer has no divisors other than 1 and itself, then it is prime.
2. **Contrapositive**: If a positive integer has divisors other than 1 and itself, then it is not prime.