Is the following statement a good definition? Why? An integer is divisible by 100 if and only if its last two digits are zeros. No, it is not. When the definition is separated into two conditional statements, one of the statements is false. No, it is not. When the definition is separated into two conditional statements, both of the statements are false. Yes, it is. When the definition is separated into two conditional statements, one of the statements is true. Yes, it is. When the definition is separated into two conditional statements, both of the statements are true.
The correct answer is: No, it is not. When the definition is separated into two conditional statements, both of the statements are false.
The statement "An integer is divisible by 100 if and only if its last two digits are zeros" is not a good definition because it is not accurate. While it is true that all integers with last two digits as zeros are divisible by 100, there are other integers with different last two digits that are also divisible by 100 (e.g. 200, 300, etc.). Thus, the statement does not cover all possible cases and is therefore incomplete.
The correct answer is: No, it is not. When the definition is separated into two conditional statements, one of the statements is false.
The statement is not a good definition because when separated into two conditional statements, only one of the statements is true. An integer is divisible by 100 if its last two digits are zeros, but it is not necessarily true that all integers with the last two digits being zeros are divisible by 100.
To determine whether the given statement is a good definition, we need to analyze the accuracy and completeness of its two conditional statements.
The statement "An integer is divisible by 100 if and only if its last two digits are zeros" can be divided into two separate conditional statements:
1. "If an integer is divisible by 100, then its last two digits are zeros." (Forward implication)
2. "If an integer's last two digits are zeros, then it is divisible by 100." (Reverse implication)
Now, let's evaluate each of these conditional statements:
1. "If an integer is divisible by 100, then its last two digits are zeros." (Forward implication)
This statement is true. If an integer is divisible by 100, it means it is a multiple of 100. By definition, any multiple of 100 must have its last two digits as zeros.
2. "If an integer's last two digits are zeros, then it is divisible by 100." (Reverse implication)
This statement is also true. If an integer has its last two digits as zeros, it means it is a multiple of 10 (as any integer that ends in zero is a multiple of 10). Therefore, it will also be a multiple of 100, as 10 is a factor of 100.
Since both conditional statements are true, the given statement is a good definition. It accurately describes the condition for an integer to be divisible by 100.
Therefore, the correct answer is: Yes, it is. When the definition is separated into two conditional statements, both of the statements are true.