2009 AMC 10A Problem 9

P1-10~2 minPrint

Sequences & Series Primes Set up the equation/formula and compute; no special trick needed

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

Show solution without answering

Key insight

The ratio squared must divide 2009 = 7 x 7 x 41, so the ratio is 7 and a = 2009 / 49 = 41.

Solution

1. Call the whole-number ratio $r$ . The terms are $a$ , then $a \times r$ , then $a \times r \times r$ . Since the terms get bigger, $r$ is at least $2$ .
2. The third term is $2009$ , so
$a \times r^2 = 2009.$
3. Break $2009$ into primes: $2009 = 7 \times 287 = 7 \times 7 \times 41$ .
4. So $r^2$ must be a perfect square that divides $7 \times 7 \times 41$ . The only such square bigger than $1$ is $7 \times 7 = 49$ . So $r = 7$ .
5. Then $a = 2009 \div 49 = 41$ .
6. Check: $41$ , $41 \times 7 = 287$ , $287 \times 7 = 2009$ . The terms increase and are whole numbers.

The answer is $\boxed{\textbf{(B)}\ 41}$ .

Why this works

With a whole-number ratio, the third term is the first term times a perfect square. So the question becomes "which perfect squares divide $2009$ ?", and the prime factorization answers it right away. The year of the contest shows up often on the AMC, so it pays to know how it factors.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Let the common ratio be the integer $r$ . Since the terms increase, $r \ge 2$ . The terms are $a$ , $ar$ , $ar^2$ , and the third equals $2009$ :
$a r^2 = 2009.$

Factor: $2009 = 7 \cdot 287 = 7 \cdot 7 \cdot 41 = 7^2 \cdot 41$ .

For $ar^2 = 2009$ with $r \ge 2$ , the perfect square $r^2$ must divide $7^2 \cdot 41$ . The only square divisor greater than $1$ is $49$ , so $r = 7$ and
$a = \frac{2009}{49} = 41.$

The sequence is $41, 287, 2009$ , which checks.

The answer is $\boxed{\textbf{(B)}\ 41}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 21: Sequences & Series (§21.1, §21.2, §21.3, §21.4)
Introduction to Number Theory, Chapter 2: Primes and Composites
Intermediate Algebra, Chapter 17: More Sequences and Series (§17.1, §17.2)
Introduction to Number Theory, Chapter 4: Prime Factorization (§4.2, §4.4)