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Question Number 110587 by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{A}\mathrm{palindrome}\mathrm{is}\mathrm{a}\mathrm{number}\mathrm{that}$$$$\mathrm{remains}\mathrm{the}\mathrm{same}\mathrm{when}\mathrm{its}\mathrm{numbers}$$$$\mathrm{are}\mathrm{reversed}.\mathrm{The}\mathrm{number}\mathrm{n}\mathrm{and}$$$$\mathrm{n}+192\mathrm{are}\mathrm{three}-\mathrm{digit}\mathrm{and}$$$$\mathrm{four}-\mathrm{digit}\mathrm{palindromes}\mathrm{respectively}.$$$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{digits}\mathrm{of}\mathrm{m}?$$$$$$$$\mathrm{Can}\mathrm{this}\mathrm{be}\mathrm{solved}\mathrm{mathematically}?$$

Answered by floor(10²Eta[1]) last updated on 29/Aug/20

$$\mathrm{we}\mathrm{just}\mathrm{need}\mathrm{to}\mathrm{get}\mathrm{a}4\mathrm{digit}\mathrm{palindrome}$$$$\mathrm{that}\mathrm{when}\mathrm{subtracted}\mathrm{by}192\mathrm{we}\mathrm{get}\mathrm{a}$$$$\mathrm{palindrome}\mathrm{with}3\mathrm{digits}$$$$\mathrm{the}\mathrm{first}4\mathrm{digits}\mathrm{palindromes}\mathrm{are}:$$$$1001$$$$1111$$$$1221$$$$....$$$$\mathrm{notice}\mathrm{that}\mathrm{we}\mathrm{can}\mathrm{stop}\mathrm{here}\mathrm{because}\mathrm{if}\mathrm{we}$$$$\mathrm{subtract}192\mathrm{of}1221\mathrm{we}\mathrm{dont}\mathrm{get}\mathrm{a}$$$$3\mathrm{digit}\mathrm{number}.$$$$\mathrm{now}\mathrm{we}\mathrm{just}\mathrm{need}\mathrm{to}\mathrm{test}\mathrm{those}2\mathrm{cases}:$$$$1001-192=809.\mathrm{not}\mathrm{a}\mathrm{palindrome}.$$$$1111-192=919.\mathrm{is}\mathrm{a}\mathrm{palindrome}$$$$\mathrm{so}\mathrm{n}=919\mathrm{and}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{digits}$$$$\mathrm{is}9+1+9=19$$

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

$$\mathrm{Thanks}.$$

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