An automorphic number reappears at the end of its square. Disregarding 0 and 1, the only one-digit automorphic numbers are 5 and 6 since 5² = 25, and 6² = 36. Find all two-digit and three-digit automorphic numbers.
Some thoughts
let a be the automorphic number exceeding 9, then a*(a-1) is divisible by 100.
Therefore, a Xor a-1 is divisible by 2.
Therefore, a Xor a-1 is divisible by 4
Similarly
a or a-1 is divisible by 25.
Now a little trial and error:
a=25 works
49,50,51 don't divide by 4
a=76 works
let a be the automorphic number exceeding 99, then a*(a-1) is divisible by 1000.
Therefore, a Xor a-1 is divisible by 8.
Similarly
a or a-1 is divisible by 125.
A little more trial and error turns up two more automorphic numbers.