- #1

Math100

- 783

- 220

- Homework Statement
- Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons in such a way that each man receives 3 bushels, each woman 2 bushels, and each child 1/2 bushel. How many men, women, and children are there?

- Relevant Equations
- None.

Proof: Let x be the number of men, y be the number of women

and z be the number of children.

We need to find the solutions in the non-negative integers

for the Diophantine equation 3x+2y+0.5z=100 such that

x+y+z=100.

From x+y+z=100, we have that z=100-x-y.

Substituting z=100-x-y into the Diophantine equation

3x+2y+0.5z=100 and multiplying it by 2 produces:

5x+3y=100.

Applying the Euclidean Algorithm produces:

5=1(3)+2

3=1(2)+1

2=2(1)+0.

Now we have that gcd(5, 3)=1.

Note that 1##\mid##100.

Since 1##\mid##100, it follows that the Diophantine equation

5x+3y=100 can be solved.

Then we have 1=3-1(2)

=3-1(5-1(3))

=2(3)-1(5)

And now I'm stuck on this problem. I know I need to find the x0 and y0 in order to seek the general solution of the Diophantine equation. How should I go from here?

and z be the number of children.

We need to find the solutions in the non-negative integers

for the Diophantine equation 3x+2y+0.5z=100 such that

x+y+z=100.

From x+y+z=100, we have that z=100-x-y.

Substituting z=100-x-y into the Diophantine equation

3x+2y+0.5z=100 and multiplying it by 2 produces:

5x+3y=100.

Applying the Euclidean Algorithm produces:

5=1(3)+2

3=1(2)+1

2=2(1)+0.

Now we have that gcd(5, 3)=1.

Note that 1##\mid##100.

Since 1##\mid##100, it follows that the Diophantine equation

5x+3y=100 can be solved.

Then we have 1=3-1(2)

=3-1(5-1(3))

=2(3)-1(5)

And now I'm stuck on this problem. I know I need to find the x0 and y0 in order to seek the general solution of the Diophantine equation. How should I go from here?

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