Triangle circumference puzzle

Christina chose a certain odd natural number divisible by three. Jacob and David then examined triangles with a perimeter in millimeters equal to the number selected by Christina and whose sides have lengths in millimeters expressed by different integers.
Jakub found a triangle in which the longest side had the longest possible length and wrote this value on the board. David found a triangle in which the shortest side has the longest possible length and wrote this value on the board. Christina correctly added both measurements on the board and got 1,681 mm.
Determine which number Christina chose.

Final Answer:

K =  2019

Step-by-step explanation:

$$\begin{gathered} K=3k,k\in N \\ K=a+b+c \\ a<b<c \\ 3k=a+b+c \\ \max =(K-1)/2 \\ min=1 \\ K=j_1+j_2+j_3 \\ {j}_1<j_2<j_3 \\ {j}_1=1\vee j_1=2 \\ {j}_2=(K-3)/2 \\ {j}_3=(K-1)/2 \\ 1+(K-3)/2+(K-1)/2=K \\ K=d_1+d_2+d_3 \\ {d}_1<d_2<d_3 \\ {d}_1=K/3-1 \\ {d}_2=K/3+0 \\ {d}_3=K/3+1 \\ {d}_1+j_3=1681 \\ K/3-1+(K-1)/2=1681 \\ 3k/3-1+(3k-1)/2=1681 \\ 3\cdot k/3-1+(3\cdot k-1)/2=1681 \\ 15k=10095 \\ k=\frac{10095}{15}=673 \\ k=673 \\ K=3\cdot k=3\cdot 673=2019 \\ {j}_1=2\text{mm} \\ {j}_2=(K-3)/2=(2019-3)/2=1008\text{mm} \\ {j}_3=(K-1)/2=(2019-1)/2=1009\text{mm} \\ {o}_1=j_1+j_2+j_3=2+1008+1009=2019\text{mm} \\ {d}_1=K/3-1=2019/3-1=672\text{mm} \\ {d}_2=K/3=2019/3=673\text{mm} \\ {d}_3=K/3+1=2019/3+1=674\text{mm} \\ {o}_2=d_1+d_2+d_3=672+673+674=2019\text{mm} \\ {o}_1=o_2=K \\ {x}_1=d_1+j_3=672+1009=1681 \\ K=2019 \end{gathered}$$

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