1. **Problem Statement:** Find all palindrome numbers less than 100, determine if there is a common factor for all these palindrome numbers, and classify these numbers as rational or irrational. 2. **Definition:** A palindrome number reads the same forwards and backwards. 3. **Step 1: List all palindrome numbers less than 100.** - Single-digit numbers (1 to 9) are palindromes: $1,2,3,4,5,6,7,8,9$ - Two-digit palindromes have the form $\overline{aa}$ where $a$ is a digit from 1 to 9: $$11,22,33,44,55,66,77,88,99$$ 4. **Step 2: Check for a common factor of all palindrome numbers less than 100.** - The palindrome numbers are: $1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99$ - The greatest common divisor (GCD) of all these numbers is 1 because 1 divides all numbers and no larger number divides all of them. 5. **Step 3: Determine if palindrome numbers are rational or irrational.** - All palindrome numbers are integers. - All integers are rational numbers because they can be expressed as a fraction $\frac{n}{1}$ where $n$ is an integer. - Therefore, all palindrome numbers are rational. **Final answers:** - Palindrome numbers less than 100: $1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99$ - Common factor of all palindrome numbers less than 100: $1$ - Palindrome numbers are rational numbers.