As Long As 128 Divides My Square

Relevant wiki: Modular Arithmetic - Problem Solving - Basic

For a number to be $0 \mod 128$ , $128$ must divide that number. For $128$ to divide a number, the number must have $2^7$ in its prime factorization. Note that squaring a number does not change its prime factorization, it simply doubles the powers of the primes. Therefore, $n$ must include $2^4$ in its prime factorization. Other solutions will simply be multiples of $2^4$ , as $128$ has no primes other than $2$ in its prime factorization. Taking the sum of all multiples of $2^4$ that are $<128$ :

$2^4(1+2+3+4+5+6+7)$

$2^4(28)$

$\boxed{448}$