Each positive integer
is subjected to the following procedure, yielding the number
:
(a) The last digit of
is moved to the first position. The resulting number is called
.
(b) The number
is squared. The resulting number is called
.
(c) The first digit of
is moved to the last position. The resulting number is called
.
(All numbers are considered in the decimal system.) For instance,
gives
,
and
.
Find all integers a such that
.
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![d = d\left(a\right)](/media/m/7/a/5/7a58ea990dcb1637ae082de0b34d413a.png)
(a) The last digit of
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
(b) The number
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
(c) The first digit of
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
![d](/media/m/f/7/d/f7d3dcc684965febe6006946a72e0cd3.png)
(All numbers are considered in the decimal system.) For instance,
![a = 2003](/media/m/b/0/4/b04aa561ac6932e382eb1bffde436eb6.png)
![b = 3200](/media/m/0/7/5/075b7a75ff76fa5676c9ebedae54e092.png)
![c = 10240000](/media/m/d/e/2/de2b2df6bec80f9415d276a6af912c0c.png)
![d = 02400001 = 2400001 = d\left(2003\right)](/media/m/d/1/e/d1edada1183f117831eca6c9e20b8093.png)
Find all integers a such that
![d\left( a\right) =a^2](/media/m/1/2/3/123f4cb019d52deeff64472735755614.png)