MHB What Is the Sum of the Digits of 3n Given These Conditions?

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The problem involves finding the sum of the digits of 3n, given that the sum of the digits of n is 100 and the sum of the digits of 44n is 800. The solution process includes analyzing the properties of digit sums and their behavior under multiplication. A correct solution has been provided by a participant named kaliprasad, indicating that the problem has been successfully tackled. The discussion encourages engagement with the Problem of the Week format and highlights the importance of following guidelines for submissions. The focus remains on mathematical reasoning and problem-solving techniques.
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Here is this week's POTW:

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A number $n$ has sum of digits 100, while $44n$ has sum of digits 800. Find the sum of the digits of $3n$.

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Congratulatons to kaliprasad for his correct solution!(Cool)

Here is a suggested solution by other:
Suppose $n$ has digits $a_1a_2\cdots a_k$ and digit sum $s=\sum a_i$. If we temporarily allow digits larger than 9, then $4n=(4a_1)(4a_2)\cdots(4a_k)$. Each carry of one reduces the digit sum by 9, so after making all necessary carries, the digit sum for $4n$ is at most $4s$. It can only be $4s$ iff there are no carries. Similarly, $11n$ has digit sum at most $2s$, with equality iff there are no carries.

Since $44n$ has digit sum $8s$, there cannot be any carries. In particular, there are no carriers in forming $4n$, so each digit of $n$ is at most 2. Hence, there are no carries in forming $3n$ and the digit sum of $3n$ is 300.
 
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