Henry Ernest Dudeney/Modern Puzzles/74 - Queer Division/Solution

by : $74$

 * Queer Division

Solution

 * $35 \, 641 \, 667 \, 749$

Proof
We are to solve the following system of simultaneous linear congruences:

First note that $(3)$ implies $(1)$.

Now we split $(2)$ using the Chinese Remainder Theorem.

We have:

Note that $(5)$ is implied by $(4)$, so we just need to simutaneously solve $(3), (4), (6)$.

As $227, 4545$ and $45454$ are pairwise coprime, the Chinese Remainder Theorem guarantees a unique solution modulo $227 \times 4545 \times 45454 = 46 \, 895 \, 573 \, 610$.

To find this solution, we first need to find, in each modulo, the inverse of the product of the other modulos.

One way to find them is to use the Euclidean Algorithm.

We have:

Next we multiply each inverse with the remainder in that modulo from our equations, and then to the product of the other modulos and sum them up.

This will give us the solution modulo $46 \, 895 \, 573 \, 610$:

which is the smallest positive solution to our congruences.