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Question Number 215272 by Rasheed.Sindhi last updated on 02/Jan/25

$$\{\begin{array}{l}\overline{abac}={\left(\overline{dc}\right)}^2\\ d=\frac{\overline{ab}}{c}\\ c^2=\overline{ac}\end{array}$$$$\overline{abac}=?$$

Answered by A5T last updated on 02/Jan/25

$${\mathrm{c}}^2=\overline{\mathrm{ac}}\Rightarrow \mathrm{c}=0,1,5\mathrm{or}6;\mathrm{d}=\frac{\overline{\mathrm{ab}}}{\mathrm{c}}\Rightarrow \mathrm{c}\ne 0$$$$\mathrm{c}=1\Rightarrow \mathrm{a}=0\Rightarrow \mathrm{d}=\mathrm{b}$$$$\Rightarrow 100\mathrm{b}+1={(10\mathrm{d}+1)}^2\Rightarrow 100\mathrm{b}+1={100\mathrm{b}}^2+1+20\mathrm{b}$$$$\Rightarrow {100\mathrm{b}}^2-80\mathrm{b}=0\Rightarrow \mathrm{b}=0\Rightarrow \mathrm{d}=0$$$$\mathrm{a}=\mathrm{b}=\mathrm{d}=0;\mathrm{c}=1\Rightarrow \overline{\mathrm{abac}}=0001$$$$$$$$\mathrm{c}=5\Rightarrow \mathrm{a}=2\Rightarrow 5\mathrm{d}=20+\mathrm{b}\Rightarrow (\mathrm{d},\mathrm{b})=(4,0);(5,5)$$$$\mathrm{Checking}\Rightarrow (\mathrm{d},\mathrm{b})=(4,0)\Rightarrow \overline{\mathrm{abac}}=2025$$$$$$$$\mathrm{c}=6\Rightarrow \mathrm{a}=3\Rightarrow 6\mathrm{d}=30+\mathrm{b}\Rightarrow (\mathrm{d},\mathrm{b})=(5,0),(6,6)$$$$\mathrm{Checking}\Rightarrow \mathrm{None}\mathrm{works}$$$$$$$$\Rightarrow \overline{\mathrm{abac}}=2025\mathrm{or}0001$$

Commented by Rasheed.Sindhi last updated on 02/Jan/25

$$\mathcal{H}appy2025sir!$$$$\mathcal{T}hanksforniceanswer!$$

Commented by A5T last updated on 02/Jan/25

$$\mathrm{Same}\mathrm{to}\mathrm{you},\mathrm{thanks}.$$

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