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Question Number 124694 by mathocean1 last updated on 05/Dec/20

$$Determinatemsuchthat$$$$miswrittenabccainbase5andis$$$$writtenbbabinbase8.$$

Answered by floor(10²Eta[1]) last updated on 05/Dec/20

$$\mathrm{M}={\mathrm{abcca}}_5={\mathrm{bbab}}_8;1⩽\mathrm{a}⩽4,0⩽\mathrm{b},\mathrm{c}⩽4$$$$\mathrm{a}.5^4+\mathrm{b}.5^3+\mathrm{c}.5^2+\mathrm{c}.5+\mathrm{a}=\mathrm{b}.8^3+\mathrm{b}.8^2+\mathrm{a}.8+\mathrm{b}$$$$\Rightarrow 309\mathrm{a}-226\mathrm{b}+15\mathrm{c}=0$$$$309\mathrm{a}-226\mathrm{b}=-15\mathrm{c}<0$$$$309\mathrm{a}<226\mathrm{b}\Rightarrow \mathrm{a}<\mathrm{b}$$$$309\mathrm{a}-226\mathrm{b}+15\mathrm{c}\equiv 4\mathrm{a}-\mathrm{b}\equiv 0\left(\mathrm{mod}5\right)$$$$4\mathrm{a}\equiv \mathrm{b}\left(\mathrm{mod}5\right)\Rightarrow (\mathrm{a},\mathrm{b})=(2,3),(1,4)$$$$\mathrm{checking}\mathrm{you}\mathrm{see}\mathrm{that}(\mathrm{a},\mathrm{b})=(2,3)\Rightarrow \mathrm{c}=4$$$$\mathrm{M}={23442}_5={3323}_8$$$$$$

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