<s>[INST] $a+b+c+d+e=abcde$ What is $\max(a)?$ The question from my nephew is this:
If $a, b, c, d, e \in N^+$ and $a+b+c+d+e=a\times b\times c\times d\times e$, then what's the the maximum possible value of $a$? Thanks ahead:)
 [/INST] $5$. This is clearly possible with $b=2$, $c=d=e=1$. It remains to show that $b+c+d+e< 6(bcde-1)$ or $6bcde>b+c+d+e+6$. However, we cannot have $b=c=d=e=1$, so $bcde\ge 2$ and, thereby, $3bcde\ge 6$. Thus we are left with $3bcde>b+c+d+e$. Assuming that $b$ is the largest number, we have $b\ge 2$, so $\frac 32bcde\ge 3cde\ge c+d+e$. Finally, $\frac32bcde>b$.
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