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Question Number 22472 by Tinkutara last updated on 18/Oct/17

$$\mathrm{If}{x}^x\cdot y^y\cdot z^z=x^y\cdot y^z\cdot z^x=x^z\cdot y^x\cdot z^y\mathrm{such}$$$$\mathrm{that}x,y\mathrm{and}z\mathrm{are}\mathrm{positive}\mathrm{integers}$$$$\mathrm{greater}\mathrm{than}1,\mathrm{then}\mathrm{which}\mathrm{of}\mathrm{the}$$$$\mathrm{following}\mathrm{cannot}\mathrm{be}\mathrm{true}\mathrm{for}\mathrm{any}\mathrm{of}\mathrm{the}$$$$\mathrm{possible}\mathrm{value}\mathrm{of}x,y\mathrm{and}z?$$$$\left(1\right)xyz=27$$$$\left(2\right)xyz=1728$$$$\left(3\right)x+y+z=32$$$$\left(4\right)x+y+z=12$$

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