Find a square number of the form aabb when written in base 10.

Let N = aabb = 1000a + 100a + 10b + b = 1100a + 11b, with a and b in the range 0..9, and at least one of them greater than 0.

Modulo 4, N = 1100a + 11b = 3b, and since N is a square which must leave a remainder of 0 or 1 when divided by 4, we have b = 0, 3, 4, 7 or 8.

Modulo 5, N = 1100a + 11b = b, and since N is a square which must leave a remainder of 0, 1 or 4 when divided by 5, we have b = 0, 1, 4, 5, 6 or 9.

By combining these two results, we have b = 0 or 4.

Also note that N = 1100a + 11b = 11(100a + b). For N to be a square, 100a + b must be a multiple of 11. Hence, modulo 11, 0 = 100a + b = a + b.

But a is a single digit, and b = 0 or 4, so the only possibility is b = 4 and a = 7.

Thus N = 7744 = 88^{2}.