- #1

Math100

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- 206

- Homework Statement
- I bought sparrows at ## 3 ## for a penny, turtle doves at ## 2 ## for a penny, and doves at ## 2 ## pence each. If I spent ## 30 ## pence buying ## 30 ## birds and bought at least one of each kind of bird, how many birds of each kind did I buy?

- Relevant Equations
- Diophantine equation: ## ax+by=c ## where ## a, b, c ## are integers.

Let ## x ## denote the number of sparrows, ## y ## denote the number of turtle doves and ## z ## denote the number of doves.

Then we have ## \frac{1}{3}x+\frac{1}{2}y+2z=30 ## such that ## x+y+z=30 ##.

Observe that

\begin{align*}

&\frac{1}{3}x+\frac{1}{2}y+2(30-x-y)=30\\

&\frac{1}{3}x+\frac{1}{2}y+60-2x-2y=30\\

&-\frac{5}{3}x-\frac{3}{2}y=-30\\

&-10x-9y=-180\\

&10x+9y=180.\\

\end{align*}

Consider the Diophantine equation ## 10x+9y=180 ##.

By the Euclidean Algorithm, we have that ## gcd(10, 9)=1 ##.

Since ## 1\mid 180 ##, it follows that the Diophantine equation ## 10x+9y=180 ## can be solved.

From ## 1=10-1(9) ##, we get ## 180=180[10-1(9)]=180(10)-180(9) ##.

Thus ## x_{0}=-180, y_{0}=180 ##.

All solutions in the integers are determined by:

## x=-180+(\frac{10}{1})t=-180+10t ## for some ## t\in\mathbb{Z} ##,

## y=180-(\frac{9}{1})t=180-9t ## for some ## t\in\mathbb{Z} ##.

Thus, ## x=-180+10t ## and ## y=180-9t ##.

To find all solutions in the positive integers of the Diophantine equation ## 10x+9y=180 ##,

we solve the following inequalities for ## t ##:

## -180+10t\geq 0 ## and ## 180-9t\geq 0 ##.

Now we have ## 18\leq t\leq 20 ##, or ## t=19 ##.

This implies ## x=10, y=9 ##, and ## z=30-10-9=11 ##.

Therefore, ten sparrows, nine turtle doves and eleven doves were purchased.

Then we have ## \frac{1}{3}x+\frac{1}{2}y+2z=30 ## such that ## x+y+z=30 ##.

Observe that

\begin{align*}

&\frac{1}{3}x+\frac{1}{2}y+2(30-x-y)=30\\

&\frac{1}{3}x+\frac{1}{2}y+60-2x-2y=30\\

&-\frac{5}{3}x-\frac{3}{2}y=-30\\

&-10x-9y=-180\\

&10x+9y=180.\\

\end{align*}

Consider the Diophantine equation ## 10x+9y=180 ##.

By the Euclidean Algorithm, we have that ## gcd(10, 9)=1 ##.

Since ## 1\mid 180 ##, it follows that the Diophantine equation ## 10x+9y=180 ## can be solved.

From ## 1=10-1(9) ##, we get ## 180=180[10-1(9)]=180(10)-180(9) ##.

Thus ## x_{0}=-180, y_{0}=180 ##.

All solutions in the integers are determined by:

## x=-180+(\frac{10}{1})t=-180+10t ## for some ## t\in\mathbb{Z} ##,

## y=180-(\frac{9}{1})t=180-9t ## for some ## t\in\mathbb{Z} ##.

Thus, ## x=-180+10t ## and ## y=180-9t ##.

To find all solutions in the positive integers of the Diophantine equation ## 10x+9y=180 ##,

we solve the following inequalities for ## t ##:

## -180+10t\geq 0 ## and ## 180-9t\geq 0 ##.

Now we have ## 18\leq t\leq 20 ##, or ## t=19 ##.

This implies ## x=10, y=9 ##, and ## z=30-10-9=11 ##.

Therefore, ten sparrows, nine turtle doves and eleven doves were purchased.