2009 AMC 12B Problem 22

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.