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Let $a $ and $ b $ be integers and consider the following subset of $ \mathbb{Z}. $ $$ a+b\mathbb{Z} = \{ a+bz \mid z \in \mathbb{Z} \}. $$ What is the intersection of $ a+b \mathbb{Z} $ and $ c+d \mathbb{Z} $ for integers $ a,b,c,d. $ Justify your argument and prove that the set $ \{ a + b \mathbb{Z}\mid a,b \in \mathbb{Z}\} $ form a basis of $ \mathbb{Z}. $

What I have done so far: I think that the intersection should be of the form $ x + bd \mathbb{Z} $ but not sure how to find $ x. $

Also would appreciate if someone could give me a hint for the basis part.

Bernard
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pseudogeek
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  • Well, that isn't correct. For example, what if $a=1,c=2,b=d=3$? Your guess is true when $\gcd(b,d)=1$ by the Chinese remainder Theorem. – Rushabh Mehta Oct 18 '21 at 14:23
  • It would probably help to work through some simple cases. If you know a little group theory, you can think of these subsets of $\mathbb Z$ as cosets of cyclic subgroups, but it isn't necessary to solving the problem. Consider the possibilities when $b=d=2$. – hardmath Oct 18 '21 at 14:24

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