Q:

Rules of Divisibility

$112,263,147,00a$ is divisible by $18.$  Find the possible values of $a.$  
To solve this I factored $18$ into $2$ and $9$, then I used the rule of $2$ and the rule of $9$.
By the rule of $2$, a may equal $0,2,4,6,8$
By the rule of $9$ we sum the digits and see if the result is divisible by $9.$  Summing the digits I have $27+a$, so a may equal $0$ or $9$
So, the only digit that follows both the rule of $2$ and $9$ is $0$.
However, when checking this with a calculator, we see that both $0$ and $9$ would work for this number to be divisible by $18.$  What have I done wrong?

A:

(to remove question from unanswered queue)
You have done nothing wrong except trusted technology too much.  The calculator you used has a limited memory when it comes to performing calculations and so incorrectly displayed $112263147009\div 18$ as an integer.
Of course, $112263147009$ being odd could not have an even number as a divisor.  Your analytical solution was indeed correct and $0$ is the only choice for the value of $a$.

