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inn [45]
3 years ago
6

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.
Mathematics
1 answer:
antiseptic1488 [7]3 years ago
7 0

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.

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