Problem 3

A boy claims that he can multiply any three-digit number by 1001 instantly. If his classmate says to him “715” he gives the answer inmediately. Compute this answer and explain the boy’s secret.

\begin{aligned} 715 \\
1001 \\
715 \\
715000 \\
715715 \\
\end{aligned}

Whenever we multiply a number by 1, we get the whole number. And we can find a pattern here, for example:

$$75 \cdot 101 = 7575$$

$$753 \cdot 1001 = 753753$$

$$7 \cdot 11 = 77$$

$$7 \cdot 10001 = 70007$$

If we have 001 * 123, we get 123, but if we have a 1001, we get 123123. As we have two “1"s.

$$1212 * 10001 = 12121212$$

So in the case of the boy. Multiplying a number by 1001 it just means writing the number twice. But this only works for a three digit numbers. In the case of a two digit number he would have to add a zero in the middle $$77 * 1001 = 77077$$. Which, whether you want it or not, it’s more effort.