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Who is Nauseated though? 

Nope. I am not, but I am interested in this XD

1)\(n^{37}\equiv c\pmod {527}\)

   We first find the factors of 527

   527 = 17 x 31 according to this calc.

   We get an equation: \(37d = 1 \pmod{480}\)

   The smallest d that satisfies this equation is 37 x 13 = 1 (mod 480)

   Note: 37 x 13 = 481

   d = 13 ;D

2) x = 148 Same answer as Guest #1

3) 1 x 2 x 3 x 4 x 5 = 2^3 x 3 x 5

    2 x 3 x 4 x 5 x 6 = 2^4 x 3^2 x 5

    3 x 4 x 5 x 6 x 7 = 2^3 x 3^2 x 5 x 7  

They both have a common factor of 2^3 x 3 x 5 = 120.

Here's a simple example. Suppose we have six pizzas and we divide each into thirds. How many slices do we have??

6 ÷ 1/3  =

6 * 3  =

18

This makes sense. Each pizza contains three slices, so 6 * 3 = 18

Remember........any positive integer divided by a fraction that is greater than 0 but less than 1 always results in an answer that is larger than that positive integer.

Are you the lieutenant of "Nauseated", or his protege "GingerAle"???

Thanks Melody. I do realize that you can get the answer by using a good calculator, such as Web2.0, HP scientific calculator, Wolfram/Alpha.....etc. What I was wondering about is this: Is there a manual rule or method that one can use to determine the remainder without resorting to sophisticated calculators?. These calculators must surly use some algorithm to arrive at the answer so quickly. They, almost certainly, do not use long division to determine the remainder.

Hi Max,

Maybe you would like to explain to our guest WHY this is so.

People learn to divide by fractons by rote - a huge number of them have no idea what they are really doing.  :)

6/(1/3)

= 6 x 3

= 18

Since the highest power on x in both the numerator and denominator is the same, as x increases without bound in both directions.......the function  will  approach the ratio of the coefficients on both these powers of x.....so  3/1 = 3

So....the function approaches y = 3 in both directions

See the graph here : https://www.desmos.com/calculator/rc5bgtop7n

Example: Make 0.25 a fraction.

\(0.25\\ =\dfrac{25}{100}\leftarrow\text{Because this is 25 hundredths}\\ = \dfrac{5}{20}\\ =\dfrac{1}{4}\)

Well you could use the web2 calculator.

mod(10^1000,348989) = 117048

Hi Xin Hu:

LOL I have found a sense of humor on you!! Please approve me to move on to Lesson 3!! (I hadn't do Lesson 1 or 2 yet XD)

(PS: Are you from China? Me from Hong Kong!! :D)

-3x - (x + 4)

= -3x - x - 4

= -4x - 4 or -4(x+1)

We can use the tangent to solve this

tan (2°)  = drop/40

40*tan (2°)  = drop = about 1.397 ft 

Actually......Melody andI have an advanced degree in Humorology...........and only one hour to do something smart???.......that's a marked improvement..........it usually takes us two or three......depending on the CDD quotient for that particular day.........

More information needed - I thought the owl said it all   LOL

Also, watch this video by Dr. James Grime explaining how to encrypt and decrypt messages using RSA system: https://www.youtube.com/watch?v=M7kEpw1tn50

You want the gradient of

\( x^2+3y=0\\ y=\frac{-x^2}{3}\)

This is a parabola so it has a different slope at every point.

The gradient of thge tangent at any given point is given by      

\(y'(x)=\frac{-2x}{3}\)

atan (260/180)

Put it into the calc.

I think that you need to be a bit more expicit with your question :)

I'd say you should try the pythagorean theorem to get the third side, and then there is a formula where you use two sides to get an angle or similar.

Thanks Alan :)