The grammar of the original question is correct. A gerund (-ing form) is used after the preposition "after". Your re-written question is also grammatically correct.
Example 1
After finishing his breakfast, Mr Chan reads the newspaper. ==> It is Mr Chan who finishes his breakfast before he starts reading the newspaper. That is to say, the doer is Mr Chan.
Example 2
After bouncing to and fro several times, the ball came to a standstill. ==> It is the ball that bounced for a while before it stopped. The doer is the ball.
However, in the original sentence "After adding 1 to a number, it becomes the LCM of 10 and 11", the number cannot add to itself. The doer of the action cannot be the number itself.
Sorry, I hope I have not diverted the focus of the thread which is about how to coach kids to solve LCM/HCF related problems.