Think of a Number/Examples/Bachet/2

Classic Problem

 * The subject chooses a number less than $60$
 * and tells you the remainders when it is divided by $3$, $4$ and $5$, separately, not successively.


 * The original number is -- what?

Solution
Let $x$ be the number chosen.

Let $A$, $B$ and $C$ be the remainders on dividing $x$ by $3$, $4$ and $5$ respectively.

Then the original number is the remainder when dividing $40 A + 45 B + 36 C$ by $60$.

Proof
We have that:

So $40 A + 45 B + 36 C$ has the same remainder on dividing by $60$ as does $x$.

Because $x$ has been chosen to be less than $60$, the result follows.