TITLE: Finding conditions on real numbers to make their sum and difference an integer
QUESTION [0 upvotes]: I m actually working on a problem and I am stuck somehow but I will be able to move on if I can figure out the following question.
Let $a, b \in \mathbb{R}$. I need $$2a, 2b,  a+b, a-b $$ all to be integers. What are the restrictions that I can place on $a,b$? Or let me say what are the conditions that I need to place on $a,b$. Thanks.

REPLY [3 votes]: From $2a, 2b$ you need both $a$ and $b$ to be either integers or half odd integers.  Then from the sum you need both to be the same among integers and half integers.  The difference doesn't give you anything new as $a-b=(a+b)-2b$ so if both terms on the right are integers so is the one on the left.
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