Question

There were 63 equal piles of plantain fruit put together and 7 single fruits. They were divided evenly among 23 travelers. What is the number of fruits in each pile? (Hint: Consider the diophantine equation 63x + 7 = 23y.

Answer 1

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:

$63x + 7 = 23y\\63x-23y = -7$

We obtain the greatest common divisor of (63,-23).

The greatest common divisor f 63 and -23 is 1.

Therefore, there exist $\alpha,\beta$ such that

$-23\alpha+63\beta=1$

By applying extended Euclidean Algorithm, we have,

$1 = (1\times 6)+(-1\times 5)\\1 = (1\times 17)+(3\times 6)\\1 = (3\times 23)+(-4\times 17)\\1 = (-4\times 40)+(7\times 23)\\1 = (7\times 63)+(-11\times 40)\\1 = (-11\times -23)+(-4\times 63)$

$1 = (-11\times -23) + (-4\times 63)$

Multiplying -7 on both sides, we get,

$-7 = (77\times -23) + 28\times 63)$

Thus, we get,

$x = 28\\y=77$

Thus, there are 28 fruits in each pile.