UNSW MathSoc Championships Q15

Let's suppose that $a=9$ , so that $\overline{abcde}=90000+\overline{bcde}$ . Since $\overline{bcde}$ divides $\overline{abcde}$ , it also divides $\overline{abcde}-\overline{bcde}=90000$ . Therefore, a good candidate for $\overline{bcde}$ is the largest 4-digit divisor of $90000$ that is not a multiple of $10$ (so that $e\neq 0$ ). Since $90000=2^4 3^2 5^4$ , the number that satisfies this condition is $3^2 5^4 = 5625$ . The other three conditions on $\overline{abcde}$ also are satisfied: $625$ , $25$ and $5$ divide $95625$ . Therefore, the answer is $95625$ .

Mathematica

Max@Select[Range[10^4,10^5-1],And@@IntegerQ/@(#/Table[Mod[#,10^i],{i,4}])&]

95625