Question 11
Let $\langle N\rangle$ be the single–digit number obtained by repeatedly adding the digits of N until only one digit remains. For example,
$\langle25\rangle\Rightarrow\langle2+5\rangle\Rightarrow\langle7\rangle\Rightarrow7$,
$\langle861\rangle\Rightarrow\langle8+6+1\rangle\Rightarrow\langle15\rangle\Rightarrow\langle1+5\rangle\Rightarrow\langle6\rangle\Rightarrow6$, and
$\langle2025\rangle\Rightarrow\langle2+0+2+5\rangle\Rightarrow\langle9\rangle\Rightarrow9$.
What is the positive difference between the smallest four–digit N and the largest four–digit N such that $\langle\langle N\rangle\times17\rangle=\langle N\rangle-1$?
$\langle25\rangle\Rightarrow\langle2+5\rangle\Rightarrow\langle7\rangle\Rightarrow7$,
$\langle861\rangle\Rightarrow\langle8+6+1\rangle\Rightarrow\langle15\rangle\Rightarrow\langle1+5\rangle\Rightarrow\langle6\rangle\Rightarrow6$, and
$\langle2025\rangle\Rightarrow\langle2+0+2+5\rangle\Rightarrow\langle9\rangle\Rightarrow9$.
What is the positive difference between the smallest four–digit N and the largest four–digit N such that $\langle\langle N\rangle\times17\rangle=\langle N\rangle-1$?