Sample Question 23:

Seven students count from 1 to 1000 as follows:

•Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, ..., 997, 999, 1000.

•Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.

•Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.

•Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.

•Finally, George says the only number that no one else says.

What number does George say?

\(\textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998\)

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Answer Keys

Question 23: C

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Solutions

Question 23
Step 1: Consider the numbers Alice says first. They are \(1, 3, 4, 6, 7, 9, \cdots\), skipping every number that is congruent to \(2 \mod 3\).

Step 2: Barbara says the numbers Alice doesn't say, but she also skips every second number which is congruent to \(5 \mod 9\).

Step 3: Following this pattern, Candice skips every number congruent to \(14 \mod {27}\), Debbie skips every number congruent to \(41 \mod {81}\), Eliza skips every number congruent to \(122 \mod {243}\), and Fatima skips every number congruent to \(365 \mod {729}\).

Step 4: The only number that is congruent to \(365 \mod {729}\) and lies between 1 and 1000 is \(365\). Thus, this is the only number that George says.

Final Answer: \(\boxed{\textbf{(C) } 365}\).