2003 AMC 12A Problem 12

Problem 12 of 25IntermediateNumber Theory

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?

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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.

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