Henry Ernest Dudeney/Puzzles and Curious Problems/141 - The First Boomerang Puzzle/Solution

by : $141$

 * The First "Boomerang" Puzzle

Solution
Let $n$ be the number that was thought of.

Let $a$ be the remainder after dividing $n$ by $3$.

Let $b$ be the remainder after dividing $n$ by $5$.

Let $c$ be the remainder after dividing $n$ by $7$.

Then:
 * multiply $a$ by $70$
 * multiply $b$ by $21$
 * multiply $c$ by $15$

Add these up.

If you get a number over $100$, subtract $105$, and continue doing so until you get a number between $1$ and $100$.

This is the number $n$ which was first thought of.

Proof
We have:

The result follows.