# Algebra Gelfand - Problem 3

## 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.