Melody

The Queen does not like all this violence.  Camelot is a peaceful, well-ordered realm :/

That is really pretty Rosala :)

Hi MG here is a question for you.  :))

1) Tell me what area is and then...

2) Tell me the approximate area of each of these shapes.  

Here is part 4  :)))

Yep Chris, we are both brilliant     LOL   :))

Thanks Mellie for making us think.  We can always rely on you for that :)

Sorry Chris I did not realize you had already answered - I only saw your first answer.

You must have posted the other s when Iwas off doing things and I did not realize that they were there. 

Anyway, we came to the same conclusion  :/

b)

Perhaps I have the answer - using your algebra Chris :)

The factors of 20 are 1,2,4,5,10,20

Now if we are going to have the largest number of primes to make 20 factors we need the most integers greater than 1 that can multiply to give 20.

they are 2*2*5

So the answer is 3 unique primes (6 primes altogether).  This is the maximum number of prime numbers that can divide this composite number with 20 factors.

a * b * c^4      

c) So the smallest positive integer that has exactly 20 positive divisors will be

3*5*2^4

$${\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{4}}} = {\mathtt{240}}$$

Thanks Chris - this was a real joint effort.   Are you happy with the answer ?   

Chris, I do not see how you have answered the question.

(Although i did get something out of your algebra logic.)

You have found - like i did - that the product of 4 unique primes can give an answr with 16 factors.

BUT the question calls for EXACTLY 20 factors.  Do you think that such a number does not exist?  

There is an app for it but it is an online tool, sorry.