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:

Among blue cards $3, 4, 5, 6,$ red $5$ divides only $5$ and red $4$ divides only $4,$ so those pairs must sit at the ends.

Red $2$ divides only $4$ and $6,$ and red $3$ divides only $3$ and $6.$ Chaining these forces the stack $R4, B4, R2, B6, R3, B3, R1, B5, R5.$

The middle three cards are $B6, R3, B3,$ summing to $6 + 3 + 3 = 12.$

Thus, the correct answer is E.

2003 AMC 10A Problem 24

Solution:

Problem 24 in Other Years