<s>[INST] If $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, maximise $x+y$. The question is: if $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, what is the greatest possible value of $x+y$?
The book says "the greatest value of $4$ is divided by $6$, which produces a remainder of $4$.  The greatest value of $y$ is when $2$ is divided by, which produces a remainder of $2$.  Therefore, the greatest value of $x+y$ is $6$."
I think what's throwing me off is the phrase "multiple of $4$" bc it makes me think that any multiple of $4$ can be divisible by $6$ (i.e. $24/6 = 4$).  The books answer doesn't use multiples, just the $4$ and $2$, respectively.  I don't understand how this works.  Can someone please clarify?
 [/INST] Take any integer $m$ and divide it by $6$. The possible remainders are $0,1,2,3,4,5$. Therefore, if we take $4m$ and divide it by $6$ then the possible remainders are $0,4,2,0,4,2$. The maximum remainder when a multiple of $4$ is divided by $6$ is $4$. 
Similarly, if you divide an integer $n$ by $3$ the possible remainders are $0,1,2$. Therefore, if we take $2n$ and divide it by $3$ then the possible remainders are $0,2,1$. The maximum remainder when a multiple of $2$ is divided by $3$ is $2$. 
Hence the maximum of the sum of the two remainders is $4+2=6$.
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