Tricky inheritance

Algebra Level 3

There is a famous problem called 17 17 camels and 3 3 sons , where an old man owns 17 17 camels that he wishes to be shared by his three sons after his death as follows: 1 2 \frac{1}{2} for the eldest son, 1 3 \frac{1}{3} for the second, and 1 9 \frac{1}{9} for the youngest.

Since 17 17 is not divisible by either 2 , 3 , 2, 3, or 9 , 9, the three sons consult a wise man, who proposes to lend them an extra camel. Now the eldest son can get his share of 18 2 = 9 \frac{18}{2} = 9 camels, the second 18 3 = 6 , \frac{18}{3} = 6, and the youngest 18 9 = 2 , \frac{18}{9} = 2, for a total of 9 + 6 + 2 = 17 9+6+2=17 camels. The extra camel can then be returned to the wise man.

Suppose now that the father has N N camels and wants to divide the herd as follows: 1 2 \frac{1}{2} for the eldest son, 1 4 \frac{1}{4} for the second, and 1 5 \frac{1}{5} for the youngest. For which of the following values of N N can the three sons apply the same borrowing strategy to ensure that the new herd will be divided fairly and that every borrowed camel will be returned to the wise man?

93 94 95 96

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Romain Bouchard
Jan 22, 2018

Relevant wiki: Number Base - Converting to Different Bases

We want the sharing of the herd to be ( N + b ) × ( 1 2 + 1 4 + 1 5 ) = N (N+b) \times (\frac{1}{2} + \frac{1}{4} + \frac{1}{5}) = N where b b is the number of borrowed camels. Hence, b = N 19 b = \frac{N}{19} and since we can only borrow whole camels ( b N b \in \mathbb{N} ) then 19 N 19 | N , so N = 95 N=\boxed{95} .

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...