Question 21
Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d(a)$:
(i) move the last digit of a to the first position to obtain the number b;
(ii) square b to obtain the number c;
(iii) move the first digit of c to the end to obtain the number d.
(All the numbers in the problem are considered to be represented in base 10.) For example, for $a = 2003$, we get $b = 3200$, $c = 10240000$, and $d = 02400001 = 2400001 = d(2003)$.
Find all numbers $a$ for which $d(a) = a^{2}$.
(i) move the last digit of a to the first position to obtain the number b;
(ii) square b to obtain the number c;
(iii) move the first digit of c to the end to obtain the number d.
(All the numbers in the problem are considered to be represented in base 10.) For example, for $a = 2003$, we get $b = 3200$, $c = 10240000$, and $d = 02400001 = 2400001 = d(2003)$.
Find all numbers $a$ for which $d(a) = a^{2}$.