Question

damian is given the following conjecture. how can he determine whether the conjecture is true? the first and third digits of a three - digit number are the same. if the second digit is equal to the sum of the first and third digits, then the number must be divisible by 11. select the correct choice below and fill in the answer box(es) to complete your choice. (use a comma to separate answers as needed.) a. the only number(s) whose digits fit the given pattern is/are. each of these is divisible by 11, so the conjecture is true. b. the only number(s) whose digits fit the given pattern is/are. of these, only is/are divisible by 11. this means there is at least one counterexample, and so the conjecture is false. c. there are too many numbers whose digits fit the given pattern for testing each to be practical. the four smallest such numbers are. of these, only is/are divisible by 11. this means there is at least one counterexample, and so the conjecture is false. d. there are too many numbers whose digits fit the given pattern for testing each to be practical. the four smallest such numbers are. none of these is divisible by 11. this means there is at least one counterexample, and so the conjecture is false. e. there are too many numbers whose digits fit the given pattern for testing each to be practical. the four smallest such numbers are. each of these is divisible by 11. it can be assumed the

Explanation:

Step1: Represent the three - digit number

Let the first and third digits be $a$ ($1\leqslant a\leqslant9$) and the second digit be $2a$ since the second digit is the sum of the first and third digits. The three - digit number can be written as $100a + 10\times(2a)+a=100a + 20a+a = 121a$.

Step2: Check divisibility by 11

We know that $\frac{121a}{11}=11a$. Since $a$ is an integer, any three - digit number that satisfies the given pattern is divisible by 11. Also, there are many such numbers (e.g., when $a = 1$, the number is 121; when $a=2$, the number is 242; etc.). The four smallest such numbers are 121, 242, 363, 484. And $\frac{121}{11}=11$, $\frac{242}{11}=22$, $\frac{363}{11}=33$, $\frac{484}{11}=44$.

Answer:

E. There are too many numbers whose digits fit the given pattern for testing each to be practical. The four smallest such numbers are 121, 242, 363, 484. Each of these is divisible by 11. It can be assumed the conjecture is true.