Four positive integers A, B, C and D have a sum of 36. If A+2 = B-2 = C+2 = D-2, what is the value of the product ABCD?
+26388 Four positive integers \(A\), \(B\), \(C\) and \(D\) have a sum of \(36\).
If \(A+2 = B-2 = C+2 = D-2\),
what is the value of the product \(ABCD\)?
\(\begin{array}{|rcll|} \hline A+2 &=& C+2 \\ \mathbf{A} &=& \mathbf{C} \\ \hline \end{array} \begin{array}{|rcll|} \hline B-2 &=& D-2 \\ \mathbf{B} &=& \mathbf{D} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline A+B+C+D &=& 36 \\ A+B+A+B &=& 36 \\ 2A+2B &=& 36 \quad | \quad : 2 \\ A+B &=& 18 \qquad (1) \\ \hline \end{array} \begin{array}{|rcll|} \hline A+2 &=&B-2 \\ A-B &=&-4 \qquad (2) \\ \hline \end{array}\)
\(\begin{array}{|lrcll|} \hline (1)+(2): & 2A &=& 18-4 \\ & 2A &=& 14 \quad | \quad : 2 \\ &\mathbf{A} &=& \mathbf{7} \\ \hline \end{array} \begin{array}{|lrcll|} \hline (1)-(2): & 2B &=& 18+4 \\ & 2B &=& 22 \quad | \quad : 2 \\ &\mathbf{B} &=& \mathbf{11} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline ABCD &=& A^2B^2 \\ ABCD &=& 7^21^2 \\ \mathbf{ABCD} &=& \mathbf{5929} \\ \hline \end{array}\)
