HMMT 2010 T3

Call a positive integer in base $10 k$-good if we can split it into two integers y and z , such that y is all digits on the left and z is all digits on the right, and such that $y=k \cdot z$. For example, 2010 is 2 -good because we can split it into 20 and 10 and $20=2 \cdot 10.20010$ is also 2 -good, because we can split it into 20 and 010 . In addition, it is 20 -good, because we can split it into 200 and 10 . Show that there exists a 48 -good perfect square.