Melody

Yes, we used to have a huge amount of fun here.  ://

1. Solve the congruence   \(3n \equiv 2 \pmod{11}\)        , 

as a residue modulo 11. (Give an answer between 0 and 10.)

3*1=3 = 3mod11

3*2=6 = 6mod11

3*3=9 = 9mod11

3*4=12 = 1mod11

3*5=15 = 4mod11

3*6=18= 7mod11

3*7=21 = 10mod11

3*8=24 = 2mod11         The answer is 8

3*9=27 = 5mod11

3*10=30 = 8mod11

3*11=33 = 0mod11

3*12=36 = 3mod11       Patteren starts repeating here     8+11t   is the more general answer

3*13=39 = 6mod11

3*14=42 = 9mod11

It might have been quicker just to say     3*-3=-9=2(mod 11)     so the general solutions is -3+11t 

-3+11=  8

Hi Max,

It was Xin Hu that was being asked - by a different anon   :)

Nauseated was our pet troll for a long time.  He lived under out toll bridge but I think the great flood of  May '16  sent him hurling down the river never to be seen again - not yet anyway.... :(

GingerAle was Nauseated's apprentice - she still shows her colours occasionally.   :)

Yes I thought that might be the case.  :)

I did look at breaking up 348989 into its prime factors but it is a prime number already so that didn't help.

I am relatively new at modular arithemtic too.  I don't think I had ever done any of it before they started appearing on this site.  So no I don't have any great ideas either.

6 divided by 1/3

Thanks Chris,    (I had wanted Max to answer but never mind )

Here is the answer I had in mind.

6 divided by 2 =3

this is because there are  3 groups of 2 in 6       2+2+2=6

1 divided by 1/3    is   3   because there are  3 thirds in 1 whole      1/3   +   1/3    +   1/3   = 1

so

2 divided by 1/3    is   6   because there are  3 thirds in 1 whole      1/3   +   1/3    +   1/3 + 1/3   +   1/3    +   1/3 = 2

so

2 divided by 1/3    is the same answer as   2 * 3/1 = 6

so

6 divided by  1/3    has the same answer as   6   * 3/1  =  6*3 = 18    

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.  :)

Well you could use the web2 calculator.

mod(10^1000,348989) = 117048

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

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      

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