16. A positive integer $n$ is said to be good if $3 n$ is a re-ordering of the digits of $n$ when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11 .
(2 marks)對於正整數 $n$, 若在十進制表示中, 整數 $3 n$ 可從 $n$ 的數字經重新排列而得出, 則 $n$ 稱為「好數」。求一個可被 11 整除的四位好數。
16. 2475
16. Let $n$ be such a number. Since $3 n$ is divisible by 3 , the sum of the digits of $3 n$ is divisible by 3 . But the sum of the digits of $3 n$ is the same as the sum of the digits of $n$, so the sum of digits of $n$ is divisible by 3 , i.e. $n$ is divisible by 3 . As a result, $3 n$ is divisible by 9 , so the sum of digits of $3 n$ is divisible by 9 . Again, the sum of the digits of $3 n$ is the same as the sum of the digits of $n$, so the sum of digits of $n$ is divisible by 9 , i.e. $n$ is divisible by 9 .
Let $n=\overline{a b c d}$. Since $n$ is to be divisible by both 9 and $11, a+b+c+d$ is divisible by 9 and $(a+c)-(b+d)$ is divisible by 11 . Considering the parities of $a+c$ and $b+d$ we see that $a+b+c+d$ has to be equal to 18 with $a+c=b+d=9$.
The rest is largely trial and error. Noting that $a$ can be no larger than 3 , we have the possibilities $(a, c)=(1,8) ;(2,7)$ or $(3,6)$. Considering the digits, the only possibilities for $n$ are $1287,1386,1485,2079,2475,2574,3465,3762$ and 3861 . Among these, we find that only $n=2475$ works as $3 n=7425$ in this case.