Since 100=2*2*5*5, any quadratic residue of 100 is also a quadratic residue of 2, 4, 5, and 25 (the prime-power divisors of 100). Conversely, any nonresidue of 100 is a nonresidue of 2, 4, 5, or 25.
The residues of 25 are 1, 4, 6, 9, 11, 14, 16, 19, 21, 24, so every residue mod 100 is one of those plus 0, 25, 50, or 75. That knocks out 60 possibilities. All of those are also residues mod 5, (they're either 1 or 4 mod 5), so that doesn't eliminate any more. Mod 4, the residues are 0 and 1, so remove any numbers that are 2 or 3 mod 4. And that's it, because everything is a residue mod 2, so we don't rule out anything there. Whatever's left is a residue mod 100.