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Cattle Rustlers


A family of 25 cattle rustling brothers meets once a year to divide up their ill-gotten gains. Each brother brings in all the cattle he has stolen over the past year. Then the brothers evenly divide up the total herd among themselves, giving the remainder to the cook at the end of the roundup.

On the day they arrive, the cattle are divvied up, and there are 3 cows left over.

That night an argument among the brothers turns nasty, and seven of them are killed. In the morning, they re-distribute the cattle, this time having seven cows left to give to the cook at the end of the roundup.

That night, the bunkhouse collapses, killing another seven brothers.

The following morning, they re-allocate the cattle among themselves, leaving 10 cows for the cook at the end of the roundup.

On the third night, the cook poisons the remaining brothers, taking the cattle for herself

How many cattle did the cook get?

Answer:

The answer is: The cook got (at least) 4003 cows! In the first pass, each brother got 160, the second, 222, and the third, 363.

Let n denote the number of cows in the herd. Then n leaves a remainder of 3 upon division by 25, a remainder of 7 upon division by 18, and a remainder of 10 upon division by 11. A common notation for this is
(1) n=3(mod 25)
(2) n=7(mod 18)
(3) n=10(mod11)
Let's start with the first equation. It gives as possible values of n the numbers 28, 53, 78,103,... The second equation says that n is 7 more than a multiple of 18, and so, since all multiples of 18 are even, n must be odd. This allows us to eliminate half of the possibilities from our first list; in particular, n is one of the numbers 53, 103, 153,... You quickly see that 153 satisfies both conditions (1) and (3). It follows that any other solution must be 153 more than a multiple of 275 (275 = 11x25). So

n = 153 + 275k, for some positive integer k.

Now look at the second condition. It gives us that n - 7 = 146 + 275k is divisible by 18. Factoring out as many 18's as we can, we get 146 + 275k = 18(8 + 15k) + (7 + 5k). Since the left hand side is divisible by 18, then 7 + 5k must be divisible by 18, too. Now you only have to check a very few numbers. The smallest possible solution is k = 14.
Therefore, n = 153 + 275(14) = 4003.
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This page revised   March 31, 2004
 
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