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An employer needs one person who is good at maths. Unless you go deep into scientific software, 90% of all programming jobs rarely require any mathematical skills at all. You will sometimes find problems in programming contests that have clever solutions if you have mathematical skills, but often very similar problems have no such clever solutions.

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For data structures and algorithms, basic algebra, exponential and logarithmic functions, and a basic course or book on discrete mathematics like one by Kenneth Rosen, should get you warmed up.

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I assume you are familiar with algebra. You should be comfortable with the concept of functions. You should be able to recognize and categorize basic functions (e.g., linear, polynomial, logarithmic, exponential). If you are not familiar with exponents and logarithms, I'd study those too. This is because the logarithmic function is frequently encountered ...

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Yuval Filmus suggests a counterexample: $x_1 + 2x_2 = 2$, $0 \le x_1 \le 1$, $0 \le x_2 \le 1$. Here there is a feasible solution to the LP instance with $x_1=1$, namely, $x_1=1$, $x_2=1/2$. However the only feasible solution to the ILP instance is $x_1=0$, $x_2=1$.

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The program $x_1 + 2x_2 = 2$ has the integer solution $(0,1)$ and the fractional solution $(1,1/2)$ but not an integer solution of the form $(1,\cdot)$.

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