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Question Number 185728 by aba last updated on 26/Jan/23

$$(\mathrm{x}-2013)!=6!7!$$$$\mathrm{x}=??$$

Answered by mr W last updated on 26/Jan/23

$$6!7!$$$$=1\times 2\times 3\times 4\times 5\times 6\times 7!$$$$=1\times 3\times 5\times 6\times 8!$$$$=1\times 5\times 2\times 9!$$$$=10!$$$$(x-2013)!=6!7!=10!$$$$x-2013=10$$$$\Rightarrow x=2023$$

Commented by Frix last updated on 27/Jan/23

$$\mathrm{Is}\mathrm{there}\mathrm{another}\mathrm{pair}(m,n)\mathrm{with}$$$$n!(n+1)!=m!$$$$\mathrm{except}\mathrm{the}\mathrm{trivial}\mathrm{ones}?$$$$\mathrm{Or}\mathrm{maybe}n!(n+k)!=m!$$$$[m,n,k\in \mathbb{N}]$$

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