CPhill

Here's my take on (b)

As per part (a), if m has 5 distinct positive divisors, it only has one prime factor - and that to the 4th power.  Let's call this prime "p."

And m*n has 14 positive divisors....... and,  if n has six distinct positve divisors, it must have "p" as one of its prime factors as well.....and it must be to the second power......this would also mean that "n" would have a second prime as a factor....call it "q"....and it would have to be to the first power......

Proof......m*n   =   [p^4]* [p^2 * q]   =  p^6*q.......note that m has (4 + 1) = 5 positive divisors....and  n has (2 + 1)(1 + 1) = (3)(2)  = 6 positive divisors and m*n has (6 + 1)(1 + 1) = (7)(2) = 14 positive divisors...therefore....m*n is composed of two distinct primes.....

Here's (a)

If  m has exactly 5 positive divisors.....there can only be one prime factor....to see why....

Let's suppose m can be prime factored as.....a^p * b^q , ......the number of possible divisors of m is  (p +1)(q + 1)  and we assume that p and q ≥ 1.......then the number of possible divisors cannot be 5 because there are no two integers  p, q ≥ 1, such that (p +1)(q + 1) = 5 .....

Similarly, if m factored as some number of primes greater than two, the least number of possible divisors would be eight... (Example......if a, b and c were primes each to the first power, the least number of divisors would be ....  2 * 2 * 2  = 8 )

Therefore, if m has 5 positive divisors, then it can only have one prime factor - and this has to be to the 4th power......( Example  ...  2^4  = 16 ....  and it has 5 positive divisors......1, 2, 4, 8, 16......)

Notice that 40^8  =  (4 * 10)^8   =  (4^8) * (10^8).......so this will have 8 trailing zeroes

And notice that 4^8  =  2^16 

Now, notice that 75 = (3 * 5 * 5)   .....so......  (75)^18 =  (3 * 5 * 5)^18  = [ 3^18 *  5^18 *  5^18] =

[3*18 * 5^18 * 5^2 * 5*16]  =  [3^18 * 5^20 * 5^16]

So    2^16 * 75^18   =    2^16 * [ 3^18 * 5^20 * 5^16]   =  [ 3^18 * 5^20] * 2^16 * 5^16  =

[ 3^18 * 5^20]  *  (2 * 5)^16  = [ 3^18 * 5^20] * (10)^16

Notice that the first product in the bracket doesn't add any zeros at all to our calculations.......thus.....

[ 3^18 * 5^20] * (10)^16   ....has 16 trailing zeros.....

And when this is combined with the  8 trailing zeros in the first part, we get 24 trailing zeros in all......

(1^2) * (2^2) * (3^2)  * (2^2)  will give us what we need = 144

Notice that the "2^2" at the end combined with the other 2^2  will give us 16  = 4^2 

Likewise, it will produce 6^2 = 36 when combined with  the "3^2"

f(x)  = (3x -5)^7    ....the derivative of this function will give us the slope at any  x value.....so...

f ' (x)  =   [7(3x - 5)^6 ]* (3)  =  21(3x -5)^6     ....... so.......

f ' (2)  = 21(3(2) - 5)^6  =   21(6 -5)^6  =  21(1)^6   = 21

I think we should ask Mellie if she can actually spell "Diophantine"............LOL!!!!!

Heron's Formula strikes, again  !!!!!!  ........ LOL!!!

Thanks for the lollipop, Melody......I'm going to add your answer to my "Watchlist" and look at it in more detail later......!!!!

I'm not sure what you're asking, here......

She has 5/8 of a yard left for each pillow.....assuming she uses the same amount for each pillow, we just need to divide this by 2......

So ....  ( 5/8 )  /  (2)   =   (5/8)/(2/1)  ...... invert the second fraction and multiply....and we have

5/8   x   1/2   =

5/16  of a yard....and that's the amount she will use on each cover......