211. "Boomerang". I call "boomerang" one of the oldest types of arithmetic puzzles. Someone is asked to think of a number and after a series of calculations, to state the result. Upon hearing the result, the person who asked the question immediately announces the thought-of number. There are hundreds of different versions of this puzzle.

The oldest recorded example of this puzzle appears to be in the "Arithmetica" of Nicomachus, who died around 120 AD. He asks you to think of any whole number from 1 to 100 and then divide it successively by 3, 5, and 7, reporting the remainder each time. Having received this information, he immediately guesses the number you thought of.

Could the reader devise a simple method to perform this feat mentally? If not, perhaps he would be interested to learn how the ancient mathematician did it.
211. Get the remainder of the division by 3, multiply it by 70, get the remainder of the division by 5 and multiply it by 21, and get the remainder of the division by 7 and multiply it by 15. Add the results, and you will get either the number you thought of or a number that differs from the thought number by an integer multiple of 105. Thus, if the number 79 was thought of, then 1, multiplied by 70, plus 4, multiplied by 21, plus 2, multiplied by 15, will give 184. Subtract 105, and you get 79 - the number you thought of.