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Question Number 100733 by john santu last updated on 28/Jun/20

$$\mathrm{A}\mathrm{positive}\mathrm{integer}\mathrm{such}\mathrm{as}4334\mathrm{is}$$$$\mathrm{a}\mathrm{palindrome}\mathrm{if}\mathrm{it}\mathrm{reads}\mathrm{the}\mathrm{same}$$$$\mathrm{forwards}\mathrm{or}\mathrm{backwards}.\mathrm{What}\mathrm{is}$$$$\mathrm{the}\mathrm{only}\mathrm{prime}\mathrm{palindrome}\mathrm{with}\mathrm{an}$$$$\mathrm{even}\mathrm{number}\mathrm{of}\mathrm{digits}?$$

Commented by Rasheed.Sindhi last updated on 28/Jun/20

$$11$$$$Anyotherpalindromemust$$$$bedivisibleby11andhence$$$$composite.$$

Commented by 1549442205 last updated on 28/Jun/20

$$\mathrm{Thank}\mathrm{you},\mathrm{sir}.\mathrm{It}\mathrm{is}\mathrm{correct}\mathrm{perfectly}.$$

Commented by Rasheed.Sindhi last updated on 28/Jun/20

$$1331={11}^3$$$$So{it}^{\prime }snotprime.$$

Commented by 1549442205 last updated on 07/Jul/20

$$\mathrm{We}\mathrm{prove}\mathrm{the}\mathrm{general}\mathrm{clause}\mathrm{following}:$$$${}^{″}\mathrm{a}\mathrm{arbirtary}\mathrm{palindrome}\mathrm{number}\mathrm{is}$$$$\mathrm{always}\mathrm{divisible}\mathrm{by}{11}^{″}.\mathrm{Indeedly},\mathrm{a}\mathrm{palindrome}$$$$\mathrm{is}\mathrm{always}\mathrm{expressed}\mathrm{in}\mathrm{form}:$$$$\mathbf{A}=\overline{{\mathbf{a}}_1{\mathbf{a}}_2...{\mathbf{a}}_{\mathbf{n}}{\mathbf{a}}_{\mathbf{n}}{\mathbf{a}}_{\mathbf{n}-1}.....{\mathbf{a}}_1=}\overline{{\mathbf{a}}_1{\mathbf{a}}_2...{\mathbf{a}}_{\mathbf{n}}}.{10}^{\mathbf{n}}+\overline{{\mathbf{a}}_{\mathbf{n}}{\mathbf{a}}_{\mathbf{n}-1}...{\mathbf{a}}_1}$$$$={\mathrm{a}}_1({10}^{2\mathrm{n}-1}+1)+{10\mathrm{a}}_2({10}^{2\mathrm{n}-3}+1)+...$$$$+{10}^{\mathrm{n}-2}{\mathrm{a}}_{\mathrm{n}-1}.({10}^3+1)+{10}^{\mathrm{n}-1}{\mathrm{a}}_{\mathrm{n}}(10+1)$$$$\mathbf{This}\mathbf{number}\mathbf{is}\mathbf{divisible}\mathbf{by}11\mathbf{because}$$$${10}^{2\mathbf{k}+1}+1=(10+1)({10}^{2\mathbf{k}}-{10}^{2\mathbf{k}-1}+{10}^{2\mathbf{k}-2}-....+1)⋮11$$

Commented by john santu last updated on 28/Jun/20

$$\mathrm{yes}..\mathrm{right}$$

Commented by john santu last updated on 28/Jun/20

$$\mathrm{only}11$$

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