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 standard method to solve this equation is the 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 is can be reduced.

We can subtract $265$ from $h'$ and $344$ from $b'$ 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.