All Answers 359343 Answers

Awww, come on!

i know everyone has their own opinions, but mines beyond

its 17, cuz it matches with Peter17$ and Peter17$AWESOME

if everyone is going to complain my number, then we release #2 favorite: 27

cuz it also matches not only with Peter17$, but it also matches with Spencer23$

the only reason my favorite number is 17 besides its compatibility with the dollies ($) is becuz it also IS NOT COMPATIBLE WITH OTHER NUMBERS except 1 and 17, no gcfs (greatest common factors), or any common factors

btw,

dollies=$

Think of a right-angled triangle in which the adjacent side to theta is √3 and the opposite is 1 (so cot(theta) = √3/1 = √3).

The hypotenuse is then: √(3+1) = √4 = 2.

This means sin(theta) = opposite/hypotenuse = 1/2

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92 if you die before your final birthday

HI!! 

That's amazing Brittany!

Funny chat!

I wouldn't even understand that story I'm struggling to understand the first line of "The Book Thief".

I'm too young to understand such a powerful plot... And first line!

2095 - 2002 = 93

Thank you Alan,

I shall think on this :)

(x+6)(x+8)

=x(x+8)        +6(x+8)

=x*x+x*8     +6*x+6*8

= x^2+8x      +6x+48

= x^2+14x+48

-9 x (-8)=+72

Percent of interest is the amount of interest divided by the original amount (and then multiplied by 100).

($622.50 ÷ $12,450) x 100  =  5.00%

You need to supply a diagram or explain in more detail how W, T¹ and T² relate to each other and to m and A.

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A half of one quarter is one eighth, so:

$$\frac{1}{3}+\frac{1}{8}\rightarrow \frac{8+3}{3\times 8}\rightarrow \frac{11}{24}$$ 

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I am thinking of a positive number, less than 500, that is divisible by every number from 1 to 10 (inclusive), with one exception. What number am I thinking of? Prove that the answer is unique.

Mmm

well if it is divisable by 10 then it is divisable by 2 and 5 as well      10,2,5

If it is divisavle by 9 then it is divisable by 3                                     9,3

If it is divisable by 8 then it is divisable by 4 and 2 as well                8,4,2

If it is divisable by 6 then it is divisable by 3 and 2 as well

All numbers are diviable by 1

So I am probably looking for a number that is not divisable by 7 but is divisable by all the others

Lets see

10*9*8*6=4320     Mmm     that is a bit bigger than 500    

The lowest common multiple of 6 and 9 is 18

The lowest common multiple of 8 and 10 is 40

18*40=720      Still too big

18=2*3*3       and 40 = 2*2*2*5

So the lowest common multiple of 18 and 40 is  2*2*2*5*3*3 = 360

So I do believe that the number you seek is 360.

Now 360 is the smallest common multiple of 1,234,5,6,8,9 and 10  

But it is not divisable by 7.

I am convinced that this is the only number that meets all these criteria but I don't know how to set out a proof 

$$\small{
\begin{array}{l}
\text{For a positive integer }$n$ \text{, }
$\varphi(n)$ denotes the number of positive integers\\
\text{less than or equal to } $n$ \text{ that are relatively prime to } $n$.\text{ What is }$\varphi(2^{100})$?
\end{array}
}$$

$$\small{
\begin{array}{rcl}
\text{Prime Factorization: }$2^{100} &=& \textcolor[rgb]{1,0,0}{2}^{100}$\\\\
$\varphi( 2^{100} ) &=& 2^{100}\cdot (1-\frac{1}{\textcolor[rgb]{1,0,0}{2}})\\\\
\varphi(2^{100}) &=& 2^{100}\cdot\frac12\\\\
\mathbf{ \varphi(2^{100})} &\mathbf{=}& \mathbf{2^{99}}=633~825~300~114~114~700~748~351~602~688$\\
\end{array}
}$$

$$\small{
\begin{array}{l}
\text{For a positive integer }$n$ \text{, }
$\varphi(n)$ denotes the number of positive integers\\
\text{less than or equal to } $n$ \text{ that are relatively prime to } $n$.\text{ What is }$\varphi(12)$?
\end{array}
}$$

$$\small{
\begin{array}{rcl}
\text{Prime Factorization: }$12 &=& \textcolor[rgb]{1,0,0}{2}^2\cdot \textcolor[rgb]{0,150,0}{3}$\\\\
$\varphi(12) &=& 12\cdot (1-\frac{1}{\textcolor[rgb]{1,0,0}{2}})\cdot (1-\frac{1}{\textcolor[rgb]{0,150,0}{3}})\\
\varphi(12) &=& 12\cdot\frac12\cdot \frac23\\
\mathbf{ \varphi(12)} &\mathbf{=}& \mathbf{4}$\\
\end{array}
}$$

This must be of the form a/60 + b/60 + c/60 = 47/60

But since it must also be of the form 1/p + 1/q + 1/r = 47/60 then a, b and c must be factors of 60 (so that each term on the LHS cancels to leave a 1 in the numerator).

$${factor}{\left({\mathtt{60}}\right)} = {{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}$$

Clearly the 3 consecutive numbers are 3, 4 and 5, so their sum is 12.

Check

$${\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{5}}}} = {\frac{{\mathtt{47}}}{{\mathtt{60}}}} = {\mathtt{0.783\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

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0.0254 is the number of metres/inch so turn 5' 2" into inches, namely 62".

Now multiply this by 0.0254 to get 0.0254metres/inch*62inches = 1.5748 metres

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Hi Mellie

It is good to see you again :))

factor(10!-1)

$${factor}{\left({\mathtt{10}}{!}{\mathtt{\,-\,}}{\mathtt{1}}\right)} = {\mathtt{29}}{\mathtt{\,\times\,}}{\mathtt{125\,131}}$$

$${factor}{\left({\mathtt{11}}{!}\right)} = {{\mathtt{2}}}^{{\mathtt{8}}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{{\mathtt{5}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{11}}$$

These two numbers are realtively prime, the only factor that they have in common is 1

I do not understand what you are trying to show with this graph Radix   

$${\frac{{\mathtt{54}}}{{\mathtt{24}}}} = {\frac{{\mathtt{27}}}{{\mathtt{12}}}} = {\frac{{\mathtt{9}}}{{\mathtt{4}}}}$$

$${\frac{{\mathtt{54}}}{{\mathtt{24}}}} = {\frac{{\mathtt{9}}}{{\mathtt{4}}}} = {\mathtt{2.25}}$$  =  $${\frac{{\mathtt{225}}}{{\mathtt{100}}}}$$

Look here: