Recently I watched a video by Arthur Benjamin:
I was curious how we solved this part of his show, and would like to know.
Essentially at this point of the video, he asks three audience members to take the number $576$ and multiply it by a $4$ digit number. Thus the resulting product is a $6$ or $7$ digit number.
Then he asks each of the members to call out all 5 of their 6, or 6 of their 7 digits, and he will find the missing digit.
The first person calls:
$8,0,9,3,8$, and Arthur guesses $8$ as the digit he leaves out.
The second person calls:
$4,7,2,5,8,4$, and Arthur guesses $6$.
The third person calls:
$9,4,4,5,4,4$, and Arthur guesses $6$.
My question is, how exactly he knew this.
Firstly, I realize that adding each of the digits together in the product yields $36$, regardless of it being a $6$ or $7$ digit number. So then I assume that he just added all their digits and subtracted from $36$ to get their missing digit.
If this is the case, where did the number $36$ come from?
I don't think this is fully true, as something like $231\times 4412 = 1019172$, which has digits sum up to $21$.
I don't have any previous experience is number theory (if I need to know this to understand why it works)