Residue class as a subset of another residue class Im currently working on a problem on residue classes, but I dunno if words are enough to prove this or there is a mathematical way? 
Prove that $M_{10} + \{2\} \subset M_5 + \{2\}$
I transformed $M_{10} + \{2\}$ as $a = 10m + 2$ => $a = 2(5m)+2$ and $M_5 + \{2\}$ as 
$b = 5m + 2$ but I have no idea how to proceed... any help?
 A: Well, we can ... just do it.
$M_{10} + 2 = \{10k + 2|k\in \mathbb Z\}$
$M_5 + 2 = \{5k + 2|k\in \mathbb Z\}$.
Easy to prove $\{10k + 2|k \in \mathbb Z\} = \{5*(2k)+2|k\in \mathbb Z\} = \{5a+2|a\in \text{ is any even integer}\} \subset \{5k+2|k \in \mathbb Z\}$.
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...or...
If you are like me, you may have had an initial instinct when first taught not to view a "modulo class" as a multivalued set but as so quasi number we can do a modified arithmetic on.
If we view it this way.
$M_{10} +\{2\}$ is some weird notation..... to express the idea of numbers of the form...  $a\equiv 2\mod 10$
We can easily prove $a\equiv 2\mod 10\implies a\equiv 2 \mod 5$ as $a\equiv 2\mod 10\implies a = 2 + 10k$ for some integer $k$ $\implies a= 2+ 5(2k) \implies a \equiv 2 \mod 5$.
..... then some how we have to figure out what that has to doe with saying that a  a modulo class is a "subset" of another... how weird....
Well.. translated.  $M_{10} + \{2\}$ is the modulo class (set) of all numbers that are congruent to $2 \mod 10$.  If $a$ is in that set, is simply to say  that $a$ is a number that is congruent  to $2 \mod 10$.  And if $a \equiv 2\mod 10$ then $a \equiv 2 \mod 5$ is easy to prove as we did above.  So if we refer $M_{5} +\{2\}$ as a class or set then $a \equiv 2 \mod 5$ means $a$ is such an number.
In set concept, we just showed an element of one set is an element of another.
So $M_{10}+\{2\}\subset M+_{5} + \{2\}$.
Moral:  Don't be afraid to think of modulo classes as sets.  And when you do the arithmetic on them, you can think of them as numbers... if, when asked, you are aware the the actual numbers you are mucking with are representative of a class of numbers.
