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Question Number 16053 by vpawarksp@gmail.com last updated on 17/Jun/17

$$numberofpositiveintegers\mathit{a}\mathit{a}\mathit{n}\mathit{d}\mathit{b}\mathit{a}\mathit{n}\mathit{d}\mathit{c}\mathit{s}\mathit{a}\mathit{t}\mathit{i}\mathit{s}\mathit{f}\mathit{y}\mathit{i}\mathit{n}\mathit{g}$$$${\mathit{a}}^{{\mathit{b}}^{\mathit{c}}}{\mathit{b}}^{{\mathit{c}}^{\mathit{a}}}{\mathit{c}}^{{\mathit{a}}^{\mathit{b}}}=5\mathit{a}\mathit{b}\mathit{c}$$

Commented by prakash jain last updated on 17/Jun/17

$$\mathrm{Given}a,b,c\mathrm{are}\mathrm{integers}$$$$\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{of}a,b,c\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}$$$$5.$$$$\mathrm{Let}a=5$$$$\mathrm{If}a=5,\mathrm{RHS}=25bc$$$$case1:$$$$\Rightarrow b^c=2orb=2,c=1$$$$5^2{2}^1{1}^{25}=50$$$$case2:$$$$\mathrm{if}a=5\mathrm{and}\mathrm{RHS}\mathrm{is}\mathrm{a}\mathrm{muliple}\mathrm{of}25$$$$\mathrm{say}b=5$$$$a^{b^c}\mathrm{increases}\mathrm{at}\mathrm{much}\mathrm{faster}\mathrm{rate}$$$$\mathrm{with}c\mathrm{than}125c.$$$$sonosolutionswitha=5,b=5$$$$-----$$$$similarlyforb=5andc=5$$$$solutions$$$$a=5,b=2,c=1$$$$b=5,c=2,a=1$$$$c=5,a=2,b=1$$

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