Test the conditional statement and its converse to determine whether the following biconditional is true. A number is divisible by 6 if and only if it is divisible by 3. It is false because at least one part of the biconditional is false. It is false because both parts of the biconditional are false. It is true because at least one part of the biconditional is true. It is true because both parts of the biconditional are true.
The conditional statement in this case is "if a number is divisible by 6, then it is divisible by 3" and its converse is "if a number is divisible by 3, then it is divisible by 6."
To test the biconditional, we need to verify if both the conditional statement and its converse are true.
1) If a number is divisible by 6, then it is divisible by 3.
This statement is true. If a number is divisible by 6, it means it can be divided evenly by both 2 and 3. So, it is divisible by 3.
2) If a number is divisible by 3, then it is divisible by 6.
This statement is also true. If a number is divisible by 3, it means it can be divided evenly by 3. It can also be divided evenly by 2 because any number divisible by 3 is also divisible by 2.
Since both the conditional statement and its converse are true, the biconditional statement "A number is divisible by 6 if and only if it is divisible by 3" is true. Hence, the correct answer is: It is true because both parts of the biconditional are true.
To test the conditional statement, we need to break it down into two parts:
1. A number is divisible by 6 only if it is divisible by 3.
2. A number is divisible by 3 only if it is divisible by 6.
Let's evaluate each part:
1. If a number is divisible by 6, is it also divisible by 3? This statement is true, as any number divisible by 6 must also be divisible by its factors, which include 3.
2. If a number is divisible by 3, is it also divisible by 6? This statement is false, as there are numbers divisible by 3 that are not divisible by 6, such as 9.
Since the second part of the biconditional is false, we can conclude that the biconditional statement, "A number is divisible by 6 if and only if it is divisible by 3," is false.
To test the biconditional statement "A number is divisible by 6 if and only if it is divisible by 3", we will test the conditional statement (if part) and its converse (only if part).
The original biconditional statement can be written as:
"If a number is divisible by 6, then it is divisible by 3, and if a number is divisible by 3, then it is divisible by 6."
Testing the conditional statement:
1. Let's take the number 12.
- If the number is divisible by 6, then it should also be divisible by 3. (True)
- However, if the number is divisible by 3, it doesn't necessarily mean it is divisible by 6. (False)
Testing the converse (or contrapositive) of the conditional statement:
2. Let's take the number 9.
- If the number is divisible by 3, then it should also be divisible by 6. (False)
- However, if the number is divisible by 6, it doesn't necessarily mean it is divisible by 3. (True)
From our tests, we can see that the conditional statement is true (1 is true, 1 is false). However, the converse of the conditional statement is false (1 is true, 1 is false).
Since the biconditional statement is only true when both the conditional and the converse are true, in this case, the biconditional statement is false.