A Question on Number System Consider the Set $A$ of natural numbers $n$ whose unit digit is non zero , such that if this units digit is erased , then the resulting number divides $n$. If $K$ is the number of elements in the set $A$ , then show that $K<25$ .
I found some two digit numbers like $22,24,26,...$ and so on but not able to show that there is no such 3 , 4 , 5 digit number.
Please help me out with this 

Let's first prove that there are no three-digit numbers (with their unit digit non-zero)with  this property. Consider a three digit number $\overline{abc}$ with $c\neq 0$. After erasing the last digit of this number the resulting number is $\overline{ab}$. Now  $$\overline{ab}\mid \overline{abc} \iff \overline{ab}\mid c$$ but this is not true as $\overline{ab}\ge10$ and $1\le c\le 9$.Therefore there are no three digit numbers ,whose unit digit is nonzero, with this property. Similarly you can prove that there are no $n\ge4$ digit numbers with this property. So to count the number of element of  set $A$ it is enough to count the number of one and two digit numbers with this property.
