Imagine following: you ask your fellow to pick any number they like, and later multiply it by your number. Once they have the result, you will ask them to circle one digit in their answer (except zero). Finally, you ask them to tell you digits that are left in any order, and you will be able to tell the circled digit.

This is simply done using some basic concepts of numerical roots. Once you know these, you will be able to construct different kinds of tricks, based on this principle.

Numerical root: A numerical root is calculated by summing up the digits in a number. Say we have 85732, so the numerical root will be 8+5 which gives you 13. Then, because 13 is bigger then 10, you take 1+3 to get 4, and add it to 7 (the third digit). So, 4+7 = 11, hence 1+1=2. Now 2+3=5, and 5+2=7. So, the numerical root of 85732 is 7. (if you know modulo arithmetic, you simply take the sum modulo 9, i.e. 8+5+7+3+2 mod 9 = 7, however, the described method is handy because it does not require that much calculations)

Interesting feature: If the numerical root is equal to 9, you can perform either addition and/or multiplication of any random number, and the numerical root will still stay the same. This is interesting because you can let your fellow to add or multiply your number with any number, any amount of times, and still know that the numerical root is 9. Let’s see how this works:

If I pick a number with a numerical root of 9, i.e. 54, and you pick 9876 (which I obviously do not know), and later you multiply these two together, and get 533304. I ask you to circle a number, you choose 5, and I ask you for the rest of the digits, i.e. 33304 (note, the order does not matter here). So, I know that the numerical root stays the same, so I find it for your digits, 3+3=6, 6+3=9, 9+4=13, 1+3=4 (as you might already see, this result is obtained when we subtract 9 from 13). So, the numerical root is 4, and I need to add 5 to get 9, hence 5 is the circled digit!

Reference:

Matematicheskie chudesa i tajny, M. Gardner, published in 1986, Moscow. (work in translation)