Henry Ernest Dudeney/Puzzles and Curious Problems/176 - Counting the Loss/Solution

by : $176$

 * Counting the Loss
 * An officer explained that the force to which he belonged originally consisted of $1000$ men, but that it lost heavily in an engagement,
 * and the survivors surrendered and were marched down to a concentration camp.


 * On the first day's march one-sixth of the survivors escaped;
 * on the second day one-eighth of the remainder escaped, and one man died;
 * on the third day's march one-fourth of the remainder escaped.
 * Arrived in camp, the rest were set to work in four equal gangs.


 * How many had been killed in the engagement?

Proof
Let $n$ be the number of survivors of the engagement.

Let $n_1$, $n_2$ and $n_3$ be the numbers left in the captured group at the end of days $1$ to $3$ respectively.

We have:

This leads us to the linear Diophantine equation:
 * $35 n - 256 k = 48$

We use Solution of Linear Diophantine Equation.

Using the Euclidean Algorithm:

Hence we see that $\gcd \set {35, -256} = 1$ which trivially divides $48$, and so there exists a solution.

Again with the Euclidean Algorithm:

From Solution of Linear Diophantine Equation, the general solution is:


 * $\forall t \in \Z: n = -5616 + 256 t, k = -768 - 35 t$