Q13/123) Are there any repeating digit whole numbers that are perfect squares?

that is

Is there any number N that satisfies all of the following conditions?

(A) N is a whole number greater than 9.
(B) N is made of a single digit that repeats (1,111 or 777,777, e.g.)
(C) N is a perfect square.

Solution

The squares mod 10 can be ending with digits 1,4,9,6,5,0
so the numbers with all 2,3,7,8 cannot be perfect square

11 mod 4 , 55 mod 4, 99 mod 4 = 3, 66 mod 4 = 2

so the numbers with all 1,5,9,6

as numbers ending with all 4s ( all 11’s * 4) cannot be a perfect square

so there is no N.