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exactly four stable endings of a number of length n

Source: 2017 Latvia BW TST P15

December 18, 2022
number theoryDigits

Problem Statement

Let's call the number string D=dn−1dn−2...d0D = d_{n-1}d_{n-2}...d_0 a stable ending of a number , if for any natural number mm that ends in DD, any of its natural powers mkm^k also ends in DD. Prove that for every natural number nn there are exactly four stable endings of a number of length nn.
[hide=original wording]Ciparu virkni D=dn−1dn−2...d0D = d_{n-1}d_{n-2}...d_0 sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe mkm^k beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.
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