5.E: Exercises

Last updated
Save as PDF

Page ID: 7600

This page is a draft and is under active development.

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

Exercise $\PageIndex{1}$

Decide whether the following homogeneous Diophantine equations are solvable if so find all the solutions.

$37x-31y=0$
$4x+28y=0$
$35x+27y=0$
$ 2058x-1935y=0$

Answer: TBC.

Exercise $\PageIndex{2}$

Decide whether the following Diophantine equations are solvable if so find all the solutions.

$37x-31y=35$
$4x+28y=113$
$35x+27y=125$
$ 2058x-1935y=54$

Answer: TBC.

Exercise $\PageIndex{3 }$:

Goldfish cost $8.00 Canadian and hamsters cost $6.00 Canadian. How many of each animal can be bought for $106.00 Canadian? Let x = number of Goldfish and y = number of hamsters. *Hint* there are 4 possible solutions.

Exercise $\PageIndex{4}$:

Pencils can be purchased for 5 pesos, and erasers can be purchased for 4 pesos. How many of each can be purchased for 27 pesos? Let x = number of pencils and y = number of erasers. *Hint* there is only one possible answer.

Answer

Solution:
Let x be the number of pencils purchased and y be the number of erasers purchased. Thus 5x + 4y = 27.
Step 1: The gcd(5, 4) was found to be 1.
Step 2: 1 | 27, so we will continue.
Step 3: One solution is $x=\frac{27}{1}(1)$ and $y=\frac{27}{1}(-1)$.
Check: 5(27) + 4(-27) = 27 $\checkmark$.
Step 4: Let u = x - 27 and v = y + 27.
Check: 0 = 5u + 5y
= 5(x - 27) + 4(y + 27)
= 5x + 4y - [5(27) - 4(27)]
= 5x + 4y - 27. $\checkmark$
Step 5: The general solutions to 5u + 4y = 0 are u = -4m and v = 5m, $m \in \mathbb{Z}$.
Step 6: x - 27 = -4m and y + 27 = 5m, $m \in \mathbb{Z}$.
Hence x = -4m + 27 and y = 5m -27, $m \in \mathbb{Z}$.

Both $x, y \geq 0$. Thus -4m + 27 $\geq$ 0 and 5m - 27 $\geq$ 0.
Solving for m in both equations we obtain m $\leq$ 6.75 and m $\geq$ 5.2.
Since $m \in \mathbb{Z}$, m = 6.

Hence x = -4(6) + 27 = 3 and y = 5(6) - 27 = 3.
Check: 5(3) + 4(3) = 15 + 12 = 27. $\checkmark$

Thus with 27 pesos, we can purchase 3 pencils and 3 erasers.

Exercise $\PageIndex{5}$:

What combination of quarters and dimes will add to a sum of 88 cents?

Exercise $\PageIndex{6}$:

When Mary cashed her check, a teller accidentally gave her as many cents as she should have dollars and as many dollars as she should have cents. Mary left with the cash adding to the money $\$27.60$ in her bag, without noticing the discrepancy. It was after she spent one half of her money, she saw that now she had the exact amount of money on the check. What was the amount of check?

Exercise $\PageIndex{7}$:

Mary bought several sweaters at $\$143$ each and bought several tops at $\$ 105$ each. She spent $ \$1500.$ How many sweaters and how many tops did she buy?

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3 }\):

Exercise \(\PageIndex{4}\):

Exercise \(\PageIndex{5}\):

Exercise \(\PageIndex{6}\):

Exercise \(\PageIndex{7}\):