What are some great mathematics tricks you know of?

This trick was taught to me by my father when I was a kid. I don’t know how it works, only that it does. If someone here knows the mechanism of how it works please let me know.

My father found this trick in a book of letters by the Lubavitcher Rebbe (Igros Kodesh vol. 8 page 266), who in turn was quoting a 15th century Talmudic work called the Kol Bo.

The Kol-Bo brings it as a trick Yeshiva students used to in order to amuse themselves (this was before the days when cat gifs were but a mouse-click away), as a way “to find a person’s age through logic [without being told explicitly]”, though the trick can work for any number.

Here’s how it goes:

You ask someone to choose a number between 1 and 100. You then ask them to divide the number into 3 and give you the remainder (e.g. if the number was 10, 9 divides neatly into three and then the remainder is 1). You then ask them to divide their original number into 5 and give you the remainder of that, and then do the same with 7.

You should now have 3 numbers, the remainders of dividing the original number into 3, 5 and 7; let’s call them x3, x5 and x7 respectively. Multiply these numbers as follows: x3 should be multiplied by 70, x5 by 21 and x7 by 15. Add them all up and if it adds up to more than 100 (technically, 105) subtract 105 until you get the right number.

An example: Let’s take the number 32.

x3 = (32 % 3) = 2
x5 = (32 % 5) = 2
x7 = (32 % 7) = 4

so now let’s multiply them:

x3 * 70 = 140
x5 * 21 = 42
x7 * 15 = 60

Adding them up we get:

140 + 42 + 60 = 242

Subtracting 105 we get 137, still more than 105, so we subtract 105 again and get 32, our original number!

This trick can work for numbers larger than 100 as well but you have to know in which group of 100 (technically 105) the original number is in. In the case of larger numbers you might have to ADD 105 instead of subtracting in order to get to the right number.

I wrote a small Ruby program to run this trick (here: Igros Rubygem), and a blog post about it (here: program[0] and program[0] 2.0).

Again, if anyone knows how this trick works please post in the comments.

Edit: thanks Gerwin Dox for explaining in the comments and Abhishek Khare for offering the following explanation:

Source: QUORA