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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?

Aug 7, 2021

#1
+26213
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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 $$ABCD$$?

$$\begin{array}{|rcll|} \hline A+2 &=& C+2 \\ \mathbf{A} &=& \mathbf{C} \\ \hline \end{array} \begin{array}{|rcll|} \hline B-2 &=& D-2 \\ \mathbf{B} &=& \mathbf{D} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline A+B+C+D &=& 36 \\ A+B+A+B &=& 36 \\ 2A+2B &=& 36 \quad | \quad : 2 \\ A+B &=& 18 \qquad (1) \\ \hline \end{array} \begin{array}{|rcll|} \hline A+2 &=&B-2 \\ A-B &=&-4 \qquad (2) \\ \hline \end{array}$$

$$\begin{array}{|lrcll|} \hline (1)+(2): & 2A &=& 18-4 \\ & 2A &=& 14 \quad | \quad : 2 \\ &\mathbf{A} &=& \mathbf{7} \\ \hline \end{array} \begin{array}{|lrcll|} \hline (1)-(2): & 2B &=& 18+4 \\ & 2B &=& 22 \quad | \quad : 2 \\ &\mathbf{B} &=& \mathbf{11} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline ABCD &=& A^2B^2 \\ ABCD &=& 7^21^2 \\ \mathbf{ABCD} &=& \mathbf{5929} \\ \hline \end{array}$$

Aug 7, 2021
#2
+26213
+2

Typo:

$$\begin{array}{|rcll|} \hline ABCD &=& A^2B^2 \\ ABCD &=& 7^211^2 \\ \mathbf{ABCD} &=& \mathbf{5929} \\ \hline \end{array}$$

heureka  Aug 7, 2021