Question

Parallelogram $ABCD$ has area $1{,}000{,}000.$ Vertex $A$ is at $(0, 0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k \gt 1,$ respectively. How many such parallelograms are there?

$49$

$720$

$784$

$2009$

$2048$

Solution:

Let $B = (b, b)$ and $D = (d, kd)$ with $b, d, k$ positive integers and $k \gt 1.$ The area is $(k - 1)bd = 1{,}000{,}000 = 2^6 \cdot 5^6.$

Each parallelogram corresponds to an ordered triple $(k - 1, b, d)$ of positive integers with product $2^6 \cdot 5^6.$ The six $2$ 's distribute among the three factors in $\binom{6 + 2}{2} = 28$ ways, and likewise the six $5$ 's in $28$ ways, giving $28^2 = 784.$

Thus, the correct answer is C.

2009 AMC 12B Problem 22

Solution:

Problem 22 in Other Years