To figure out the inverse of the conditional statement "If a number is a prime number, then it has no factors other than one and itself," let's analyze it step by step.
A conditional statement has the form:
"If P, then Q."
Here, P is "a number is a prime number," and Q is "it has no factors other than one and itself." Thus, our statement can be written as:
- If P, then Q.
The inverse of a statement is formed by negating both the hypothesis (P) and the conclusion (Q) of the original statement. Therefore, the inverse of "If P, then Q" is:
- If not P, then not Q.
Applying this to our statement:
- P: "a number is a prime number"
- Q: "it has no factors other than one and itself"
The inverse becomes:
- If a number is not a prime number (not P), then it has factors other than one and itself (not Q).
Let's match this to the options provided:
1. "If a number is not a prime number, then it has factors other than one and itself."
2. "If a number has no factors other than one and itself, then it is a prime number."
3. "A number is a prime number if and only if it has no factors other than one and itself."
4. "If a number has factors other than one and itself, then it is not a prime number."
The correct inverse statement is:
- "If a number is not a prime number, then it has factors other than one and itself."
From the options provided, the appropriate match for the inverse statement is:
4. "If a number has factors other than one and itself, then it is not a prime number."
Thus, option 4 is the correct answer.
