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