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There are 725 such multiples of 5.

There are 14400 possible sequences.

Atoms are mainly composed of protons, neutrons, electrons, and sometimes morons. Atoms that have morons and have extra neutrons are isotopes of morons and have higher density.

Generally, if more than a 10% of the atoms in your body have morons then you belong in one or more classes of morons. Those who have isotopes of morons belong in the groups of dense morons.

thankyouthankyouthankyou!! 

Thank you!

When t  = 0     we have this equation

n  +  a  =  2       (1)

When t  = 1   we have this equation 

n + m   =   3       (2)

When  t  = 2   we have this equation

n + 2m  - ab^2   = 20      (3)

When  t = 3  we have this equation

n + 3m  =  1     (4)

And  when  t  =4  we have this

n + 4m +  ab^t  =   -162     (5)

Using (2)  and (4) we have that

n + m  = 3

n  + 3m  =  1        subtract the first equation from the second and we have that

2m  =  -2

m  = -1

So....  n =  4

And using (1)

4  +  a  = 2

a  = -2

And using (3)  we have that

n+ 2m  -ab^2  = 20

Subbing we have that

4  + 2(-1)  - (-2)b^2  = 20

2  + 2b^2  = 20

1  + b^2  =  10

b^2  = 9

b =   3  or  b  = -3

Assuming that  b  is positive

So  using 5...if   n = 4, m= -1, a =-2  and  b  = 3    and  t  = 4  we have that

4 - 4  + (-2)(3)^4   =

-2 * 81  = 162

So  

n = 4

m = -1

a = -2

b = 3

Here's a graph using  x  instead of  t  : https://www.desmos.com/calculator/b677bwjlwf

Thank you.

Shortcut:

There's a certain ratio betveen the area of a golden rectangle and the area of a circle in which it's been inscribed.

That ratio is:  ≈ 1 : 1.756

So, we have:    8 x 1.756 = 14.048

6xyz + 30xy + 21xz + 2yz  + 105x + 10y + 7z  = 812

3x ( 2yz + 10y + 7z)  +  (2yz + 10y + 7z)  + 105x  =  812

3x (2yz + 10y + 7z)  + (2yz + 10y + 7z)  + 3x(35)  = 812

3x [ 2yz +10y + 7z + 35 ] + [2yz + 10y + 7z]  =  812        add  35 to both sides

3x ( 2y +7)(z + 5) +  2yz + 10y + 7z  + 35   =  812  + 35

3x  (2y + 7) (z + 5)  +  1* (2y + 7)(z +5)  =   847      factor the left side

(3x + 1) (2y + 7) (z + 5)  =   847

Note that

7 * 11^2  = 847    ....so....

  7 * 11 * 11    =  847

So we can let    x  =2, y = 2   and z  = 6

So

x + y +  z  =   2  +  2  +  6   =    10

If I understand the question:
A number is perfect square and a perfect cube if it follows this rule: If its prime factor(s) is a multiple of 6. So, you would try the first few small primes raised to power of: 6, 12, 18.......etc:

2^6 = 64 too small. 2^12 =4096 still too small. 2^18 =262144 - just right! so that one number.
3^6 =729 too small. 3^12 =531441 - just right. So that is your 2nd number.
5^6 =15625 - too small. 5^12 =244,140,625 way too large.
7^6 =117649 - just right. So, that is your 3rd number.
11^6 =1,771,561 - too large.
And that is it. You have ONLY 3 such numbers:

Number        Sq.root       Cubic root
117649           343             49
262144           512             64
531441           729             81

a=0;b=0;c=0;p=0; cycle:d=a*100+b*10+c;if(6*a*b*c+30*a*b+21*a*c+2*b*c+105*a+10*b+7*c==812, goto loop, goto next); loop:printd," ",;p=p+1; next:c++;if(c<10, goto cycle, 0);c=0;b++;if(b<10, goto cycle, 0);b=0;c=0;a++;if(a<10, goto cycle,0);print"Total = ",p

x       y      z

2      2      6

P.S. I haven't got a clue on how to solve it algebraically !!

Sorry EP: I mean the same 6-digit number is a perfect square AND a perfect cube of  2 different numbers such as 4096 which is a perfect square of 64 and a perfect cube of 16.

How can they be perfect squres and pefect cubes of the SAME number?????

  Do you mean perfect squares and perfect cubes of DIFFERENT numbers?...or something else?

    BTW...I cannot answer this Q............   

"However, the p hat symbol is used to refer to an even smaller group: a proportion of the sample of the entire group. The p hat is often used when discussing the results of surveys. For example, a statistician asks a sample group of 500 adults if they like grape soda; 342 say yes, 100 say no and 58 have no opinion. The statistician is interested in which proportion like grape soda, so an equation will be formulated to find the p hat, or the proportion that answered "yes." The p hat will then be expressed as a percentage or a fraction of the sample group."

Had to look that up....   so I think p-hat will be   6 out of 15 = .4 (or 40 percent)

...and this:

"To understand what the p hat symbol represents and how it is used, the difference between a population and a sample must first be understood. In the study of statistics, the word "population" refers to the entire group that is being studied. A "sample," on the other hand, is only a part of that group. A sample is, ideally, representative of the entire group, meaning that a statistician can study the habits, behaviors and characteristics of that small group and then generalize the findings to the entire population. In statistics, a lower-case p without a caret above it stands for "population proportion." When a statistician calculates the population proportion, the entire group is being factored into the equation."

  So I THINK p is the percentage of the entire population that likes non-fiction and is given as 42 percent or .42

I am not sure of these answers.....but I know you needed SOMEthing....maybe someone will now chime in and tell if I am correct or incorrect in my 'guess'.       (I think this part is incorrect.......will get back to ya)

Didn't you already post this before?

It is not off topic, it is mathematics.

Thanks for sharing guest.

There are many great youtube clips available.

People who are serious about leanrning maths should make good use of them.

THX, EP  and Melody.....!!!!