7. The positive integers $a, b, c, d$ satisfy the equation $a b=2 c d$.
Which of the following numbers could not be the value of the product $a b c d$ ?
A 50
B 100
C 200
D 450
E 800
Solution
B

Since $a b=2 c d$, the product $a b c d=2 c d \times c d=2(c d)^{2}$, hence it must be twice a perfect square. This is true for all the options, except 100 since $100=2 \times 50$ but 50 is not a perfect square. $\left[50=2 \times 5^{2} ; 200=2 \times 10^{2} ; 450=2 \times 15^{2} ;\right.$ and $800=2 \times 20^{2}$. $]$