heureka

Stephan is standing on a mesa at the Painted Desert. The elevation of the mesa is about 1380 meters and Stephan's eye level is 1.8 meters above ground. If Stephan can see a band of multicolored shale at the bottom and the angle of depression is 29 degrees, about how far is the band of shale from his eyes?

$$\small{\text{$
\begin{array}{rcl}
\sin{(29\ensurement{^{\circ}})} &=& \dfrac{1380~\mathrm{m}+1.8~\mathrm{m}}{s}\\\\
s &=& \dfrac{1380~\mathrm{m}+1.8~\mathrm{m}}{ \sin{(29\ensurement{^{\circ}})} }\\\\
s &=& \dfrac{ 1381.8~\mathrm{m} }{ 0.48480962025 }\\\\
\mathbf{s} & \mathbf{=} & \mathbf{2850.19 ~\mathrm{m}}
\end{array}
$}}$$

The band of shale is 2850.19 m far from his eyes.

$$\\\small{\text{
d. What is the largest positive integer $n$ such that 1754, 1457 and 368 }}\\
\small{\text{
all leave the same remainder $r$ when divided by $n$?
}}$$

I.

$$\small{\text{
In a. we habe the difference $1754-1457 = 297 $
}} \\
\small{\text{
In b. we habe the difference $1754-368= 1386 $
}} \\
\small{\text{
In c. we habe the difference $1457 -368= 1089 $
}} \\
\begin{array}{lrcl}
\end{array}
}}$$

II.

$$\small{\text{
The prime factorisation all differences:
}} \\ \\
\small{\text{$
\begin{array}{lrcl}
(1) & 297 &=& 3^3\cdot 11 \\
(2) & 1386 &=& 2\cdot 3^2\cdot 7 \cdot 11 \\
(3) & 1089 &=& 3^2 \cdot 11^2 \\
\end{array}
$}} \\$$

III.

$$\small{\text{
The greatest $n$ is the greatest common divider of the differences:
}} \\ \\
\small{\text{$
\begin{array}{lrcll}
(1) & 297 &=& 3 &\mathbf{ \cdot 3^2 \cdot 11 }\\
(2) & 1386 &=& 2\cdot 7 &\mathbf{ \cdot 3^2 \cdot 11 }\\
(3) & 1089 &=& 11 &\mathbf{ \cdot 3^2 \cdot 11 }\\
\end{array}
$}} \\
\small{\text{$
\boxed{~~
n= gcd {(297,1386,1089)}=99 ~~}
$}}$$

IV.

$$\small{\text{
The greatest $n$ is $3^2\cdot 11 = 99$. The same remainder is 71 :
}} \\ \\
\small{\text{$
\begin{array}{lrcll}
(1) & 1754 &\equiv & 71 \pmod{99} \\
(2) & 1457 &\equiv & 71 \pmod{99} \\
(3) & 368 &\equiv & 71 \pmod{99}
\end{array}
$}} \\$$

$$\\\small{\text{
c. What is the largest positive integer $n$ such that 1754 and 368 }}\\
\small{\text{
leave the same remainder $r$ when divided by $n$?
}}$$

$$\small{\text{$
\begin{array}{lrcl}
(1) & 1754 &\equiv & r \pmod n\\
(2) & 368 &\equiv & r \pmod n\\
\\
\hline
\\
(1) & 1754-r &=& a\cdot n \qquad a\in Z \\
(2) & 368-r &=& c\cdot n \qquad c\in Z \qquad c\ne a\\
\\
\hline
\\
(1) & r &=& 1754- a\cdot n \\
(2) & r &=& 368- c\cdot n \\
\\
\hline
\\
(1)=(2) & 1754- a\cdot n &=& 368- c\cdot n\\
& 1386 &=& n\cdot (a-c) \qquad | \qquad n \mathrm{~is~ max, ~when ~}(a-c) = 1\\
& \mathbf{n} & \mathbf{=} & \mathbf{1386}
\\
\hline
\\
(1) & 1754 &\equiv& 368 \pmod{1386} \\
(2) & 368 &\equiv& 368 \pmod{1386} \\
\end{array}
$}}$$

$$\\\small{\text{
b. What is the largest positive integer $n$ such that 1457 and 368 }}\\
\small{\text{
leave the same remainder $r$ when divided by $n$?
}}$$

$$\small{\text{$
\begin{array}{lrcl}
(1) & 1457 &\equiv & r \pmod n\\
(2) & 368 &\equiv & r \pmod n\\
\\
\hline
\\
(1) & 1457-r &=& b\cdot n \qquad b\in Z \\
(2) & 368-r &=& c\cdot n \qquad c\in Z \qquad c\ne b\\
\\
\hline
\\
(1) & r &=& 1457- b\cdot n \\
(2) & r &=& 368- c\cdot n \\
\\
\hline
\\
(1)=(2) & 1457- b\cdot n &=& 368- c\cdot n\\
& 1089 &=& n\cdot (b-c) \qquad | \qquad n \mathrm{~is~ max, ~when ~}(b-c) = 1\\
& \mathbf{n} & \mathbf{=} & \mathbf{1089}
\\
\hline
\\
(1) & 1457 & \equiv & 368 \pmod{1089} \\
(2) & 368 & \equiv & 368 \pmod{1089} \\
\end{array}
$}}$$

$$\\\small{\text{
a. What is the largest positive integer $n$ such that 1457 and 1754 }}\\
\small{\text{
leave the same remainder $r$ when divided by $n$?
}}$$

$$\small{\text{$
\begin{array}{lrcl}
(1) & 1754 &\equiv & r \pmod n\\
(2) & 1457 &\equiv & r \pmod n\\
\\
\hline
\\
(1) & 1754-r &=& a\cdot n \qquad a\in Z \\
(2) & 1457-r &=& b\cdot n \qquad b\in Z \qquad b\ne a\\
\\
\hline
\\
(1) & r &=& 1754- a\cdot n \\
(2) & r &=& 1457- b\cdot n \\
\\
\hline
\\
(1)=(2) & 1754- a\cdot n &=& 1457- b\cdot n\\
& 297 &=& n\cdot (a-b) \qquad | \qquad n \mathrm{~is~ max, ~when ~}(a-b) = 1\\
& \mathbf{n} & \mathbf{=} & \mathbf{297}
\\
\hline
\\
(1) & 1754 &\equiv & 269 \pmod{297} \\
(2) & 1457 &\equiv & 269 \pmod{297}
\end{array}
$}}$$

solve for x: (3^(x+2))*(5^(x-1))=15^(2x)

$$\small{\text{$
\begin{array}{rcl}
3^{x+2} \cdot 5^{x-1} &=& 15^{2x}\\
3^x\cdot 3^2 \cdot 5^x \cdot 5^{-1} &=& 15^{2x}\\
3^x\cdot 5^x \cdot 3^2 \cdot 5^{-1} &=& 15^{2x}\\
( 3\cdot 5)^x \cdot 3^2 \cdot 5^{-1} &=& 15^{2x}\\
15^x \cdot \dfrac{9}{5} &=& 15^{2x}\\
15^x \cdot 1.8 &=& 15^{2x}\\
15^x \cdot 1.8 &=& 15^{x+x}\\
15^x \cdot 1.8 &=& 15^x\cdot 15^x \qquad | \qquad : 15^x\\
1.8 &=& 15^x\\
15^x &=& 1.8 \qquad | \qquad \ln{()}\\
\ln{( 15^x )} &=& \ln{ (1.8) } \\
x\cdot \ln{( 15 )} &=& \ln{ (1.8) } \\\\
x &=& \dfrac{ \ln{ (1.8) } } { \ln{( 15 )} } \\ \\
x &=& \dfrac{ 0.58778666490 } { 2.70805020110 } \\\\
\mathbf{x} & \mathbf{=} & \mathbf{0.21705161325}
\end{array}
$}}$$

$$\\ \small{\text{
Several congruences are listed below and labeled with powers of 2.
}}\\
\small{\text{
Some of the congruences are true and some are false.
}}\\
\small{\text{
Find the sum of the labels for the congruences that are true.
}}\\$$

$$\small{\text{$
\begin{array}{rrcl|rcl|rcl|r}
(1) & 118 &\equiv & 25 \pmod{13} & 118 \pmod{13} &\equiv& 1 & 25\pmod{13} &\equiv& 12 & \mathrm{false}\\
(2) & 2401 &\equiv & 147 \pmod{49} & 2401 \pmod{49} &\equiv& 0 & 147\pmod{49} &\equiv& 0 & \mathrm{true}\\
(4) & 183 &\equiv & 291 \pmod{6} & 183 \pmod{6} &\equiv& 3 & 291\pmod{6} &\equiv& 3 & \mathrm{true}\\
(8) & 2701 &\equiv & 14393 \pmod{8} & 2701 \pmod{8} &\equiv& 5 & 14393\pmod{8} &\equiv& 1 & \mathrm{false}\\
(16)& 493 &\equiv & 873 \pmod{10} & 493 \pmod{10} &\equiv& 3 & 873\pmod{10} &\equiv& 3 & \mathrm{true}\\
(32)& 4113 &\equiv & 396 \pmod{9} & 4113 \pmod{9} &\equiv& 0 & 396\pmod{9} &\equiv& 0 & \mathrm{true}\\
\end{array}
$}}$$

sum of the labels for the conguences that are true: (2)+(4)+(16)+(32) = 54

$$\small{\text{
Find the sum of all of the integers
}}\\
\small{\text{
in the set $\{-18,27,54,254,754,1036,3254\}$
}}\\
\small{\text{
that are congruent to 6 modulo 8.
}}$$

$$\small{\text{$
\begin{array}{rcl}
-18 \equiv -2 & \equiv & 6 \pmod{8} \\
27 \equiv 3 & \not\equiv & 6 \pmod{8} \\
54 \equiv 6 & \equiv & 6 \pmod{8} \\
254 \equiv 6 & \equiv & 6 \pmod{8} \\
754 \equiv 2 & \not\equiv & 6 \pmod{8} \\
1036 \equiv 4 & \not\equiv & 6 \pmod{8} \\
3254 \equiv 6 & \equiv & 6 \pmod{8} \\
\end{array}
$}}$$

the sum is -18 + 54 + 254 + 3254 = 3544

$$\small{\text{
Find the units digit of $3^{777}$ or $3^{777} = x \pmod {10} $
}}$$

$$\small{\text{$
\begin{array}{lrcll}
1. & ~~gcd(10,3) &=& 1 \qquad & | \qquad gcd = \mathrm{greatest~common~ divisor}\\
2. & ~~\varphi{(10)} &=& 4 \qquad & | \qquad \varphi = \mathrm{Euler ~function}\qquad \varphi{(10)} = 10 \cdot (1-\dfrac{1}{2})\cdot (1-\dfrac{1}{5}) \\
3. & ~~a^{\varphi (n)} &\equiv& 1 \pmod{n} \qquad & | \qquad \mathrm{Euler's ~Theorem,~~ if~ } gcd(a,n)=1 \qquad \\
& ~~ 3^4 &\equiv& 1 \pmod {10} \qquad & | \qquad n = 10\qquad a = 3\\\\
& & 3^{777} \pmod{10}\\\\
& &\equiv& 3^{4\cdot 194 +1} \pmod{10}\\
& &\equiv& 3^{4\cdot 194}\cdot 3 \pmod{10}\\
& &\equiv& (3^4)^{194}\cdot 3 \pmod{10}\\
& &\equiv& (\underbrace{3^4}_{=1 \pmod{10}})^{194}\cdot 3 \pmod{10}\\\\
& &\equiv& 1^{194}\cdot 3 \pmod{10}\\
& &\equiv & \textcolor[rgb]{1,0,0}{3} \pmod{10}\\ \\
& \mathbf{the~ Units~ digit~ of ~} 3^{777} \mathbf{ ~is ~} 3\\
\end{array}
$ }}$$

$$\small{\text{
If $n \equiv 43 \pmod{60}$, then what is the residue of $n$ modulo 6?
}}$$

$$\small{\text{$
\begin{array}{lrcl}
(1) & n - 43 &=& 60\\
(2) & n-x &=& m\cdot 6 \\
\\
\hline
\\
(1)-(2) & n-43 - n + x &=& 60 - m\cdot 6 \\
& -43 + x &=& 60 - m\cdot 6 \\
&x &=& 60+43- m\cdot 6 \\
&x &=& 103- m\cdot 6 \qquad | \qquad m =17 \mathrm{~~lowest
~positive ~integer}\\
&x &=& 103-102\\
& \mathbf{x} & \mathbf{=} & \mathbf{1 }
\end{array}
$}}\\$$

the residue of n modulo 6 is 1