<s>[INST] $xy=22$ and $yz=26$: What is $x+y+z $ equal to? Given the following: $$xy=22,\qquad yz=26,$$ where $x,y,z\in\mathbb{N}$. Which of the following is a possible value of $ x + y + z $? 
$ \textbf {(A) } 22 \qquad \textbf {(B) } 24 \qquad \textbf {(C) } 26 \qquad \textbf {(D) } 48 $ 
 [/INST] We know that $(x+z)y=48$. We are looking for $(x+z)+y$.
$$
x+z=1;y=48\implies (x+z)+y=49\\
x+z=2;y=24\implies (x+z)+y=26\\
x+z=3;y=16\implies (x+z)+y=19\\
x+z=4;y=12\implies (x+z)+y=16\\
x+z=6;y=8\implies (x+z)+y=14\\
$$
we could switch the values, and get the other choices, but the sums and products remain the same.
The only common element is $26$.
If we want the actual values, note that $(x+z)y=2\cdot24$. Since neither $xy=22$ nor $yz=26$ are divisible by $24$, we need $y=2$. Then we get $x=11$ and $z=13$.
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