Is the following statement a good definition? Why? An integer is divisible by 100 if and only if its last two digits are zeros. Yes, it is. When the definition is separated into two conditional statements, both of the statements are true. Yes, it is. When the definition is separated into two conditional statements, one of the statements is true. 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, both of the statements are true.
The correct answer is: No, it is not. When the definition is separated into two conditional statements, one of the statements is false.
To explain why, let's break down the given definition:
"An integer is divisible by 100 if and only if its last two digits are zeros."
Breaking down the definition into two conditional statements, we have:
1. If an integer is divisible by 100, then its last two digits are zeros.
2. If an integer's last two digits are zeros, then it is divisible by 100.
The first statement is true because any integer divisible by 100 will indeed have its last two digits as zeros.
However, the second statement is false. This is because an integer with its last two digits as zeros may or may not be divisible by 100. For example, the number 5000 has its last two digits as zeros, but it is divisible by 100. However, the number 2002 also has its last two digits as zeros, but it is not divisible by 100.
Since one of the two conditional statements is false, the overall definition cannot be considered a good definition.