Answer:
There are 28 fruits in each pile.
Step-by-step explanation:
We are given the following in the question:
Let x denote the number of fruits in each pile and y denote the number of fruits that every traveler receive.
Consider the diophantine equation:
![63x + 7 = 23y\\63x-23y = -7](https://tex.z-dn.net/?f=63x%20%2B%207%20%3D%2023y%5C%5C63x-23y%20%3D%20-7)
We obtain the greatest common divisor of (63,-23).
The greatest common divisor f 63 and -23 is 1.
Therefore, there exist
such that
![-23\alpha+63\beta=1](https://tex.z-dn.net/?f=-23%5Calpha%2B63%5Cbeta%3D1)
By applying extended Euclidean Algorithm, we have,
![1 = (1\times 6)+(-1\times 5)\\1 = (1\times 17)+(3\times 6)\\1 = (3\times 23)+(-4\times 17)\\1 = (-4\times 40)+(7\times 23)\\1 = (7\times 63)+(-11\times 40)\\1 = (-11\times -23)+(-4\times 63)](https://tex.z-dn.net/?f=1%20%3D%20%281%5Ctimes%206%29%2B%28-1%5Ctimes%205%29%5C%5C1%20%3D%20%281%5Ctimes%2017%29%2B%283%5Ctimes%206%29%5C%5C1%20%3D%20%283%5Ctimes%2023%29%2B%28-4%5Ctimes%2017%29%5C%5C1%20%3D%20%28-4%5Ctimes%2040%29%2B%287%5Ctimes%2023%29%5C%5C1%20%3D%20%287%5Ctimes%2063%29%2B%28-11%5Ctimes%2040%29%5C%5C1%20%3D%20%28-11%5Ctimes%20-23%29%2B%28-4%5Ctimes%2063%29)
![1 = (-11\times -23) + (-4\times 63)](https://tex.z-dn.net/?f=1%20%3D%20%28-11%5Ctimes%20-23%29%20%2B%20%28-4%5Ctimes%2063%29)
Multiplying -7 on both sides, we get,
![-7 = (77\times -23) + 28\times 63)](https://tex.z-dn.net/?f=-7%20%3D%20%2877%5Ctimes%20-23%29%20%2B%2028%5Ctimes%2063%29)
Thus, we get,
![x = 28\\y=77](https://tex.z-dn.net/?f=x%20%3D%2028%5C%5Cy%3D77)
Thus, there are 28 fruits in each pile.