Let n be an integer value. Which of the following expressions evaluates to true if and only if n is a two-digit integer (i.e., in the range from 10 to 99, inclusive)?
Understand the Problem
The question is asking which logical expression accurately determines if an integer n is a two-digit number between 10 and 99, inclusive.
Answer
B: $(n \geq 10) \text{ AND } (n < 100)$
Answer for screen readers
The correct answer is B: $(n \geq 10) \text{ AND } (n < 100)$.
Steps to Solve
- Understand the range for two-digit numbers
Two-digit numbers fall between 10 and 99, inclusive. This means: 10 ≤ $n$ ≤ 99.
- Evaluate option A
Check if the expression $n = (n \text{ MOD } 100)$ is true for two-digit numbers.
- This expression means that $n$ is equal to its last two digits.
- Example: If $n = 57$, then $n \text{ MOD } 100 = 57$.
- This can be true for values of $n$ greater than or equal to 0, so it does not specifically validate 10 to 99.
- Evaluate option B
Check if the expression $(n \geq 10) \text{ AND } (n < 100)$ is true.
- This checks if $n$ is at least 10 and less than 100, which correctly represents the range for two-digit numbers.
- Evaluate option C
Check the expression $(n < 10) \text{ AND } (n \geq 100)$.
- This will never be true for any integer, as there are no integers that are both less than 10 and greater than or equal to 100.
- Evaluate option D
Check the expression $(n > 10) \text{ AND } (n < 99)$.
- This expression checks for numbers strictly greater than 10 and strictly less than 99.
- While it includes some two-digit numbers, it excludes 10 and 99, so it does not fully represent the range.
The correct answer is B: $(n \geq 10) \text{ AND } (n < 100)$.
More Information
This logical condition accurately captures all integers from 10 to 99. It combines two checks: one ensuring that $n$ is not less than 10, and the other ensuring it is not 100 or greater, thus covering the full range of two-digit integers.
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