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\(\frac{3x+12}{x+4} = \frac{4x^2 + 8x + 8x^2}{2x}\\ \frac{3(x+4)}{x+4} = \frac{12x^2 + 8x }{2x}\\ \frac{3(x+4)}{x+4} = \frac{4x(3x + 2 )}{2x}\\ 3= 2(3x + 2) \\ \frac{3}{2}= 3x + 2 \\ -\frac{1}{2}=3x\\ x=-\frac{1}{6}\)

I don't really understand the wording of the question but if it means

\(xy - \frac{x}{y^2} - \frac{y}{x^2} = 3.\\~\\ and\\~\\ (x-1)(y-1)=k \)

Find all possible values of k  then this graph indicates that there are no possible values of k.

2021202120212021  mod 17 ==0

[1^2009 + 2^2009 + 3^2009 + ... + 100^2009] mod 10^10 ==7532998500 - these are the last 10 digits.

60 ÷ (1+7) = 60 ÷ 8 = 7...... 4

{Rick take four first then Marty takes N, and Rick

takes (8-N), which can ensure that the last one is

taken by Rick.}

For winning of a game,

Must be obey the rules

of the game. i.e. last

french fries finished on

his turn.

Total french fries = 60

Rick or Morty may eat 1 to 7

french fries on a turn

Now,

        Rules for win is the last 

   french fries eat on a turn.

If rick start to eat fries

than, for wining Rick must be

eat last french fries

So, the only possible way is

      if he pick up 4 fries on his turn

     than he win i.e 60/4 = 15 times

     total turn out of the ricks turn

     8. So he win

     So, the option (b) is correct

Use the Wolfram-Alpha Theorem to get x=-1/6

:< It was part of a problem, but I guess I did the first part wrong lol

can you tell me how you did this?

Thank you so much!

The only values that work are x = 2 and y = 2, so (x - 1)(y - 1) = 1.

Let us call a position where you can win if you play correctly a winning position, and any other position a losing one. We can see that having 1-7 left is a winning position, and 8 is a losing position, etc. Notice that whatever amount \(x\) that Morty takes on a given turn, Rick can always take \(8-x\) french fries, and thus 8 will be taken in total. This means that if Rick can turn the number of french fries into a multiple of 8, Morty will be in a losing position, and so Rick will win by just taking \(8-x\) french fries each turn, lowering the number by 8 each time until it reaches 0. In order to turn 60 into a multiple of 8, you need to take away 4, so the answer should be \(\boxed{b) \space 4}\)

I will look later but it looks very tricky. ..... 

Thanks Alan  

Least positive integer n ==780

780^2 mod 1560 ==608,400 mod 1560 ==0

2^100 - 1 == 3 * 5^3 * 11 * 31 * 41 * 101 * 251 * 601 * 1801 * 4051 * 8101 * 268501

Product==3  x  5 ==15

Your number is =978264

978264 mod 7==0

978264 mod 8==0

978264 mod 9==0

The fair price is 5.83.

The center is (4,6).

x+y=1096500/529979

Percentage difference = (difference/original)x100

80-68= 12

(12/80)x100= -15% or by 0.85

Percentage difference = (difference/original)x100

80-68= 12

(12/80)x100= -15% or by 0.85

Solved simeltaneously (assuming the question), 

x=12 and y=-1