Henry Ernest Dudeney/Puzzles and Curious Problems/14 - Horses and Bullocks/Solution

by : $14$

 * Horses and Bullocks

Solution

 * $252$ horses and $327$ bullocks.

Proof
We have that:
 * $\pounds 17, 4 \shillings = 344 \shillings$
 * $\pounds 13, 5 \shillings = 265 \shillings$

Hence we are to find the solution to the Diophantine equation:


 * $344 h - 265 b = 33$

To quote :
 * This is easy enough if you know how, but we cannot go into the matter here.

If one decides to go into the matter, one finds that a standard method to solve this equation is to use the Euclidean Algorithm on $344$ and $265$:

Now we reverse the equations:

Therefore:
 * $33 = \paren {33 \times 104} \times 344 - \paren {33 \times 135} \times 265$

which gives the solutions:
 * $h' = 3432, b' = 4455$

but this solution can be reduced.

We can subtract $265$ from $h'$ and $344$ from $b'$ simultaneously to obtain smaller solutions.

We now get:
 * $h = 3432 - 12 \times 265 = 252$
 * $b = 4455 - 12 \times 344 = 327$

which is minimal, and we check that

Hence the dealer has bought $252$ horses and $327$ bullocks.