# Problem Of The Week #397 Dec 15th, 2019

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#### anemone

##### MHB POTW Director
Staff member
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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#### anemone

##### MHB POTW Director
Staff member
Congratulatons to kaliprasad for his correct solution!

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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