Mathematics

Four positive integers, A, B, C, D have a sum of 36.

If A+2=B-2=C*2=D/2.

What is A*B*C*D?

\(\begin{array}{|rcll|} \hline A+2 &=& B-2 \\ \mathbf{A} &=& \mathbf{B-4} \\ \hline \end{array} \begin{array}{|rcll|} \hline 2C &=& B-2 \\ \mathbf{C} &=& \mathbf{\dfrac{B-2}{2}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \dfrac{D}{2} &=& B-2 \\ \mathbf{D} &=& \mathbf{2(B-2)} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline A+B+C+D &=& 36 \\ (B-4)+B+\dfrac{B-2}{2}+2(B-2) &=& 36 \\ B-4+B+\dfrac{B}{2}-1+2B-4 &=& 36 \\ 4B+\dfrac{B}{2}-9 &=& 36 \\ 4B+\dfrac{B}{2} &=& 36+9 \\ 4B+\dfrac{B}{2} &=& 45 \quad | \quad \cdot 2 \\ 8B+B &=& 90 \\ 9B &=& 90 \quad | \quad : 9 \\ \mathbf{B} &=& \mathbf{10} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline \mathbf{A} &=& \mathbf{B-4} \\ A &=& 10-4 \\ \mathbf{A} &=& \mathbf{6} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{C} &=& \mathbf{\dfrac{B-2}{2}} \\ C &=& \dfrac{10-2}{2} \\ \mathbf{C} &=& \mathbf{4} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{D} &=& \mathbf{2(B-2)} \\ D &=& 2(10-2) \\ \mathbf{D} &=& \mathbf{16} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline A*B*C*D &=& 6*10*4*16 \\ \mathbf{A*B*C*D}&=&\mathbf{3840} \\ \hline \end{array}\)