Positive integers a, b, and 2009, with a < b < 2009, form a geometric sequence with an integer ratio. What is a?

2009 AMC 10A Problem 9

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Positive integers

a,

b,

and

2009,

with

a \lt b \lt 2009,

form a geometric sequence with an integer ratio. What is

a?

7

41

49

287

2009

If the ratio is

r,

then

a r^2 = 2009

Factor

2009 = 7^2 \cdot 41

to find the only integer ratio greater than

1

Let the common ratio be

r.

Then

a r^2 = 2009 = 7^2 \cdot 41.

Since

r

must be an integer greater than

1,

the only possibility is

r = 7,

giving

a = 41

and the sequence

41, 287, 2009.

Thus, the correct answer is B.