Question

Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6.$ She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

$8$

$9$

$10$

$11$

$12$

Solution:

Since $4$ divides only $4$ and $5$ divides only $5$ among $3,4,5,6,$ the two ends must be $R4,B4,\dots,B5,R5.$

Because $2$ divides only $4$ and $6,$ the next card is $R2,B6,$ and since $3$ divides only $3$ and $6,$ the full stack is $R4,B4,R2,B6,R3,B3,R1,B5,R5.$

The middle three cards are $6,3,3,$ which sum to $12.$

Thus, the correct answer is E.

2003 AMC 12A Problem 12

Solution:

Problem 12 in Other Years