SKMO

Petra had written natural numbers from 1 to 9. She added two of these numbers, deleted them, and wrote the resulting sum instead of the summaries. She thus had eight numbers written down, which she managed to divide into two groups with the same product. Find how large this product could be.

Correct answer:

x =  504

Step-by-step explanation:

$$\begin{gathered} m=9!=362880 \\ {m}_1=\sqrt{m}=\sqrt{362880}=72\sqrt{70}\doteq 602.3952 \\ 6=2\cdot 3 \\ 7=5+2=(5+9)/2 \\ 8=4\cdot 2=2\cdot 2\cdot 2 \\ 9=3\cdot 3 \\ 5+9=14=2\cdot 7 \\ A\cdot B\cdot C\cdot D=E\cdot F\cdot G\cdot (H+I) \\ 1\cdot 6\cdot 7\cdot 8=2\cdot 3\cdot 4\cdot (5+9)=336 \\ 1\cdot 8\cdot 7\cdot 9=3\cdot 6\cdot 4\cdot (2+5)=504 \\ 3\cdot 2\cdot 8\cdot 7=1\cdot 6\cdot 4\cdot (9+5)=336 \\ 3\cdot 4\cdot 6\cdot 7=1\cdot 8\cdot 9\cdot (2+5)=504 \\ 4\cdot 2\cdot 6\cdot 7=1\cdot 3\cdot 8\cdot (5+9)=336 \\ 504=2\times 2\times 2\times 3\times 3\times 7 \\ 336=2\times 2\times 2\times 2\times 3\times 7 \\ x=504 \end{gathered}$$

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