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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.

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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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    I.e., $a,b \in \{\ldots,-1.5,-0.5,0.5,1.5,\ldots\}$ or $a,b \in \{\ldots,-2,-1,0,1,2,\ldots\}$. – 2017-01-24
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    Thank you guys. One more question please. Am i allowed to call the set $$ \{....., -1.5,-0.5,0.5,1.5, ...\} =\frac{ \mathbb{Z}}{2}?$$ – 2017-01-24
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    I just realised that I am wrong. – 2017-01-24
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    I think its $$\mathbb{Z} + \frac{1}{2}$$ – 2017-01-24
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    @Jaynot: that only gets you the case $a,b$ both half odd integer. It misses the case where both are integer. David has it right. – 2017-01-24