Solution below.

Reveal solution by highlighting:

A binary number with 1 as the last digit is odd, so we only have to look at the odd ones.

101₁₀ = 1100101₂. 2 0s in a row, out. Let’s go on. My feeling is, that the solution isn’t very far away, as those 2 0s near the low end of the number, and there has to be a way to “fix them”.

103₁₀ = 1100111₂. 2 0s in a row, out.

105₁₀ = 1101001₂. 2 0s in a row, out. Original 0s fixed, but now there’s a new problem.

107₁₀ = 1101011₂. And that’s our solution.