heureka

The least common multiple of two positive integers is 7!, 

and their greatest common divisor is 9. If one of the integers is 315,

then what is the other?
 

Formula:

\(\begin{array}{|rcll|} \hline GCD(M, N) \times LCM(M, N) &=& M \times N \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline GCD(M, N) \times LCM(M, N) &=& M \times N \\ 9\times 7! &=& 315\cdot N \\\\ N &=& \dfrac{9\times 7!}{315} \\\\ &=& \dfrac{ 7!}{35} \\\\ &=& \dfrac{ 2\times 3 \times 4 \times 5 \times 6 \times 7 }{5\times 7} \\\\ &=& 2\times 3 \times 4 \times 6 \\\\ &=& 144 \\ \hline \end{array} \)

The other is 144

Find the total number of four-digit palindromes.

(Recall that a palindrome is a non-negative sequence of digits which reads the same forwards and backwards, such as 1331. Zero cannot be the first digit.)

There are 90 four-digit palindromes.

\(\begin{array}{|r|r|r|} \hline & n & \text{palindrome } a(n) \\ \hline 1. & 110& 10\ 01 \\ 2. & 111& 11\ 11 \\ 3. & 112& 12\ 21 \\ 4. & 113& 13\ 31 \\ 5. & 114& 14\ 41 \\ 6. & 115& 15\ 51 \\ 7. & 116& 16\ 61 \\ 8. & 117& 17\ 71 \\ 9. & 118& 18\ 81 \\ 10. & 119& 19\ 91 \\ \hline 11. & 120& 20\ 02 \\ 12. & 121& 21\ 12 \\ 13. & 122& 22\ 22 \\ 14. & 123& 23\ 32 \\ 15. & 124& 24\ 42 \\ 16. & 125& 25\ 52 \\ 17. & 126& 26\ 62 \\ 18. & 127& 27\ 72 \\ 19. & 128& 28\ 82 \\ 20. & 129& 29\ 92 \\ \hline 21. & 130& 30\ 03 \\ 22. & 131& 31\ 13 \\ 23. & 132& 32\ 23 \\ 24. & 133& 33\ 33 \\ 25. & 134& 34\ 43 \\ 26. & 135& 35\ 53 \\ 27. & 136& 36\ 63 \\ 28. & 137 & 37\ 73 \\ 29. & 138& 38\ 83 \\ 30. & 139& 39\ 93 \\ \hline 31. & 140& 40\ 04 \\ 32. & 141& 41\ 14 \\ 33. & 142& 42\ 24 \\ 34. & 143& 43\ 34 \\ 35. & 144& 44\ 44 \\ 36. & 145& 45\ 54 \\ 37. & 146& 46\ 64 \\ 38. & 147& 47\ 74 \\ 39. & 148& 48\ 84 \\ 40. & 149& 49\ 94 \\ \hline 41. & 150& 50\ 05 \\ 42. & 151& 51\ 15 \\ 43.& 152& 52\ 25 \\ 44. & 153& 53\ 35 \\ 45. & 154& 54\ 45 \\ 46. & 155& 55\ 55 \\ 47. & 156& 56\ 65 \\ 48. & 157& 57\ 75 \\ 49. & 158& 58\ 85 \\ 50. & 159& 59\ 95 \\ \hline 51. & 160& 60\ 06 \\ 52. & 161& 61\ 16 \\ 53. & 162& 62\ 26 \\ 54. & 163& 63\ 36 \\ 55. & 164& 64\ 46 \\ 56. & 165& 65\ 56 \\ 57. & 166& 66\ 66 \\ 58. & 167& 67\ 76 \\ 59. & 168& 68\ 86 \\ 60. & 169& 69\ 96 \\ \hline 61. & 170& 70\ 07 \\ 62. & 171& 71\ 17 \\ 63. & 172& 72\ 27 \\ 64. & 173& 73\ 37 \\ 65. & 174& 74\ 47 \\ 66. & 175& 75\ 57 \\ 67. & 176& 76\ 67 \\ 68. & 177& 77\ 77 \\ 69. & 178& 78\ 87 \\ 70. & 179& 79\ 97 \\ \hline 71. & 180& 80\ 08 \\ 72. & 181& 81\ 18 \\ 73. & 182& 82\ 28 \\ 74. & 183& 83\ 38 \\ 75. & 184& 84\ 48 \\ 76. & 185& 85\ 58 \\ 77. & 186& 86\ 68 \\ 78. & 187& 87\ 78 \\ 79. & 188& 88\ 88 \\ 80. & 189& 89\ 98 \\ \hline 81. & 190& 90\ 09 \\ 82. & 191& 91\ 19 \\ 83. & 192& 92\ 29 \\ 84. & 193& 93\ 39 \\ 85. & 194& 94\ 49 \\ 86. & 195& 95\ 59 \\ 87. & 196& 96\ 69 \\ 88. & 197& 97\ 79 \\ 89. & 198& 98\ 89 \\ 90. & 199& 99\ 99 \\ \hline \end{array}\)

If k is constant, what is the value of k so that the polynomial

\(k^2x^3-8kx+16\) is divisible by \(x-1\)?

\(k^2x^3-8kx+16 = k^2(x-1)(\ldots )(\ldots) \)

The first root is 1.

\(\begin{array}{|rcll|} \hline k^2 \cdot 1^3 - 8k \cdot 1 + 16 &=& 0 \\ k^2 - 8k + 16 &=& 0 \\ k &=& \dfrac{ 8\pm \sqrt{64-4\cdot 16} }{2} \\ k &=& \dfrac{ 8\pm 0 }{2} \\ k &=& \dfrac{ 8\pm 0 }{2} \\ \mathbf{k} & \mathbf{=} & \mathbf{4} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline 16x^3-32x+16 : (x-1) = 16x^2+16x-16 \\ \hline \end{array}\)

By what integer factor must 9! be multiplied to equal 11!?

\(\begin{array}{|rcll|} \hline 9!\times 10 \times 11 &=& 11! \\ 9! \times 110 &=& 11! \\ \hline \end{array}\)

The integer factor is 110.

Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$,

where $1 \le n \le 25$, is $f(n)$ prime?

\(\begin{array}{|r|rcr|c|} \hline n & && f(n) & \text{prime} \\ \hline 1 & 1 &=&1 & \\ 2 & 1+2&=&3 & \checkmark \\ 3 & 1+3&=&4 & \\ 4 & 1+2+4&=&7 & \checkmark \\ 5 & 1+5&=&6 & \\ 6 & 1+2+3+6&=&12 & \\ 7 & 1+7&=&8 & \\ 8 & 1+2+4+8&=&15 & \\ 9 & 1+3+9&=&13 & \checkmark \\ 10 & 1+2+5+10&=&18 & \\ 11 & 1+11 &=& 12 & \\ 12 & 1+2+3+4+6+12 &=& 28 & \\ 13 & 1+13 &=& 14 & \\ 14 & 1+2+7+14 &=& 24 & \\ 15 & 1+3+5+15 &=& 24 & \\ 16 & 1+2+4+8+16 &=& 31 & \checkmark \\ 17 & 1+17 &=& 18 & \\ 18 & 1+2+3+6+9+18 &=& 39 & \\ 19 & 1+19 &=& 20 & \\ 20 & 1+2+4+5+10+20 &=& 42 & \\ 21 & 1+3+7+21 &=& 32 & \\ 22 & 1+2+11+22 &=& 36 & \\ 23 & 1+23 &=& 24 & \\ 24 & 1+2+3+4+6+8+12+24 &=& 60 & \\ 25 & 1+5+25 &=& 31 & \checkmark \\ \hline \text{sum } & &&& 5 \\ \hline \end{array}\)

How many perfect squares have a value between 10 and 1000?

\(4^2 = 16 \ldots 31^2=961\)

\(31 - 3 = \mathbf{28}\) perfect squares have a value between 10 and 1000.

A line is described by the equation $y-4=4(x-8)$.

What is the sum of its $x$-intercept and $y$-intercept?

\(\begin{array}{|rcll|} \hline y-4 &=& 4(x-8) \\ y-4 &=& 4x-32 \\ 4x-y &=& 32-4 \\ 4x-y &=& 28 \quad & |\quad : 28 \\ \dfrac{x}{7}-\dfrac{y}{28} &=& 1 \\ \dfrac{x}{\mathbf{7}}+\dfrac{y}{\mathbf{-28}} &=& 1 \\\\ \mathbf{7-28} &\mathbf{=}& \mathbf{-21} \\ \hline \end{array} \)

The sum is -21

A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$.

What is the value of $j$?

\(\begin{array}{|lrcll|} \hline & 2x+3y&=& 21 \\\\ \text{Point$_1$} & 2\cdot 2+3\cdot (-9) &=& -23 \qquad \text{is parallel} \\ \text{Point$_2$} & 2\cdot j+3\cdot (17) &=& -23 \\ & 2\cdot j &=& -23 - 3\cdot (17) \\ & 2\cdot j &=& -23 - 51 \\ & 2\cdot j &=& -74 \\ & \mathbf{j} &\mathbf{=}& \mathbf{-37} \\ \hline \end{array} \)

If $\overline{40\blacklozenge}$ represents a three-digit positive integer with a ones digit of $\blacklozenge$ and $\overline{1\blacklozenge}$ is a two-digit positive integer with a ones digit of $\blacklozenge$, what value of $\blacklozenge$ makes the equation $\overline{40\blacklozenge} \div 27 = \overline{1\blacklozenge}$ true?

\(40{\color{red}5}\div 27 = 1{\color{red}5}\)

can you help me find the speed of the airplane?

\(\begin{array}{|rcll|} \hline \vec{P} &=& \binom{300}{0} \\ \vec{W} &=& \binom{-35\cdot \cos(75^{\circ})} {-35\cdot \sin(75^{\circ})} \\\\ \vec{R} &=& \vec{P} + \vec{W} \\ &=& \binom{300}{0} + \binom{-35\cdot \cos(75^{\circ})} {-35\cdot \sin(75^{\circ})} \\ &=& \binom{300-35\cdot \cos(75^{\circ})}{-35\cdot \sin(75^{\circ})} \\ &=& \binom{300-9.05866657859}{-33.8074039201} \\ &=& \binom{290.941333421}{-33.8074039201} \\\\ |\vec{R}| &=& \sqrt{290.941333421^2+(-33.8074039201)^2} \\ &=& \sqrt{84646.8594930+1142.94055982} \\ &=& \sqrt{85,789.8000528} \\ &=& 292.898958777 \\ \hline \end{array}\)

The resulting speed of the airplane is \( 292.9\ \dfrac{km}{h}\)