Amazing Cryptogram #2

$0.\text{TALKTALKTALKTALK}\cdots=\dfrac{\text{TALK}}{9999}$ .

Therefore $\text{DID}=101,~303,~909$ .

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$i)~\text{DID}=101$

$99\times\text{EVE}=\text{TALK}$ and since $E\geq2$ , the left side is impossible to be a four-digit number.

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$ii)~\text{DID}=303$

$33\times\text{EVE}=\text{TALK}$ .

Since $33\times414=13662>9999$ , $\text{E}=1$ or $\text{E}=2$ .

If $\text{E}=1$ , $\text{K}=3$ . But then $\text{D}=\text{K}$ , which doesn't satisfy the problem.

So $\text{E}=2$ . Since $\text{DID}$ and $\text{EVE}$ are coprime integers, there are 5 values for $\text{EVE}$ .

$33\times212=6{\color{#D61F06}{99}}6 \\ 33\times242=7986 \\ 33\times2{\color{#D61F06}{6}}2=8{\color{#D61F06}{6}}4{\color{#D61F06}{6}} \\ 33\times2{\color{#D61F06}{7}}2=89{\color{#D61F06}{7}}6 \\ 33\times292=9{\color{#D61F06}{6}}3{\color{#D61F06}{6}}$

So we know that $\text{EVE}=242$ and $\text{TALK}=7986$ .

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$iii)~\text{DID}=909$

$11\times\text{EVE}=\text{TALK}$ .

Since $\text{E}=\text{K}$ needs to be satisfied in order for the equation above to hold, it contradicts the problem.

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Therefore, the original expression was $\dfrac{242}{303}=0.7986798679867986\cdots$ .

$\therefore\text{L+E+V+I+T+A+T+E+D}=8+2+4+0+7+9+7+2+3=\boxed{42}$ .