Olympiad of May 2016

Question

We say that a four-digit number $\overline{abcd} $, which starts at $ a $ and ends at $ d $, is interchangeable if there is an integer $ n> 1 $ such that $ n * \overline {abcd} $ is a four-digit number that starts in $ d $ and ends in $ a $. For example, $ 1009 $ is interchangeable since $ 1009 * 9 = $ 9081. Find the largest interchangeable number.

Solution: (not complete)

I came to 2248 as the largest interchangeable number thru trial and error

Answer 1

Partial Answer:

Suppose that $a=4$. Then, $n=2$.

So, $4bc8*2=8ef4$. But, $8\cdot2\bmod{10}=6$, not $4$.

So, $a<4$. If $a=3$, then $n=2$ or $n=3$. If $n=3$, then

$3bc9\cdot3=9ef3$. But, $9\cdot3\bmod{10}=7\neq3$. So $n=2$, in which case

$3bc6\cdot3=6ef3$. But, $6\cdot3\bmod{10}=8\neq3$.

So, $a\leq2$.