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There are 303 possible sequences.

The product abc is -44.

GCD[17!,  29!] =17!

17! = 2^15 * 3^6 * 5^3 * 7^2 * 11 * 13 * 17 [It has 7 distinct prime factors]

LCM[17!,  29!] =29!

29! = 2^25 * 3^13 * 5^6 * 7^4 * 11^2 * 13^2 * 17 * 19 * 23 * 29 [It has 10 distinct prime factors]

a=has at most 7 distinct prime factors.

https://web2.0calc.com/questions/abcdefgh-is-a-regular-octagon-of-side-12cm

Oh crap,no I mean all the possible numbers that could equal s that: - 2(s - 7) > 4s + 8 is true

You see what I meant?

64p - 144 ≤  0  →   64p ≤ 144   →  p ≤ 144/64  →   p ≤  9/4

22 - 20p < 0  →   22 < 20p   →  22/20 < p  →   p >  11/10

p = ( 11 / 10,  9 / 4 ] 

I think your probability should read: 5 / 72  NOT  25 / 72. If that is correct, then:

[(nC2) * 5^(n - 2)] / 6^n =5 / 72, solve for n

n = 3 dice.

Thx....glad to help  !!!!

Total capacity =  pi * r^2 * h  / 3   =    pi * (16)^2 * 120  / 3 =  10240  pi ft^3

.25 capacity = 2560 pi

Using similar  triangles we  have that        r / h =  16 /120  =  2 /15  so    r =  2h  /15   

So

2560 pi  =  pi  ( 2h/15)^2  * h    simplify

2560  = (4/225)h^3

2560 (225/4)  = h^3

144000 = h^3       take the cube root of both sides

 h ≈ 52.4 ft   when  25%  of capacity

Whoa, that one tripped me up, but thank you!

2) is  a little  difficult

At the end of Week 5 of my basketball team's season, our record is 3 wins and 17 losses. Each week after that, we win 3 games and lose 1 game. At the end of which week will my team have won at least 65% of its games total for the first time in the season?

Call the  number of  additional weeks  (past week 5) that it takes to win 65% of the games,  W

Our team plays 4 games per week and we win 3 of these

So

Total games won

________________  =     .65

Total games played

3  + 3 W

____________  =   .65

(17 + 3) + 4W

3 + 3 W

________  = .65           multiply both sides  by  20 + 4W

20 + 4W

3 + 3 W  =  .65 ( 20 + 4 W )     simplify

3 + 3 W  = 13  + 2.6 W      subtract  2.6 W, 3   from both sides

.4W  = 10        divide both sides by .4

W = 25   (additional weeks past week 5 )   ....so  ...after the 30th week

Check

3 + 25(3)                   78

__________   =      ____  =  .65    !!!!

20 + 25 (4)              120

 I'm laughing at my own error.

Instead of   ∠DCG = ∠CBG = arctan(2 / 1)    should be  ∠DCG = ∠CBG = arctan(1 / 0.5)

I'll assume  that this is   10t / { 25 + t ]

So

3.75  = 10t  / [ 25 + t]        multiply both sides by  25 + t

3.75 [ 25 + t]  = 10t      simplify

93.75  + 3.75 t =10 t      subtract 3.75 t from both sides

93.75  = 6.25 t     divide both sides by 6.25

t  =15  (min)

BE and CF intersect at 90º

∠DCG = ∠CBG = arctan(2 / 1)

CG = sin(∠CBG * BC)

Use the law of cosines to calculate the length of line DG.

If   FG / GH   = 1......then G is the  midpoint of FH

So....to find the  x coordinate of  G we have  that  [ 6a + a ]  / 2 =   (7/2)a 

All the possible  numbers  are those from  - infinity  up to, bur not including 1

We obviously cannot list every value

(2) is  missing  something

" At the end of which week will my team have won at least ???? of its games total for the first time in the season? "

A couple

6)  hrs in a week  =  24 * 7  = 168......can you  take it from here??

5)    -4x > -24                   divide both sides by -4 and  reverse the inequality  sign

       x <  6   →   (-inf, 6)  

4)   2x ≥  -10      divide both sides by 2

       x ≥  -10/2

      x ≥  -5   →  [ -5, inf )

b^2 - a^2  =72

(b - a) ( b + a)  = 72

(19 -17) (19 + 17)  = 72

(2)  ( 36)  = 72

a = 17   b =19    c =21

c^2 - b^2  =  21^2  - 19^2  =  (21 - 19) (21 + 19)  = (2) (40)  =   80

Thanks, that actually makes sense I wasnt thinking about it the right way 🤦

But what I mean is find all the possible numbers that could equal s, rereading the question may help.

-2 ( s - 7)  >  4s + 8

-2s + 14  > 4s + 8

6 > 6s

1 > s

Or we can read this as

s < 1

(-inf, 1 )

Let

D = (0, 0)

C = (1,0)

E = (0, 1/2)

F = (1/2, 1)

B = (1,1)

Slope of line connecting EB = [ 1 -1/2 / [ 1 - 0 ] =   1/2

Equation of lime  connecting EB    ....y = (1/2) x + 1/2      [ 1 ] 

Slope of line connecting CF   =  [ 1 - 0 ] / [1/2 -1 ] = 1 / -1/2   =  -2 

Equation of line  connecting CF  ......y = -2 ( x - 1) ..... y = -2x + 2    [2]

x intersection [1] and [2]  = x value of G

(1/2)x + 1/2  = -2x + 2

(5/2)x = 3/2

5x = 3

x = 3/5

And using either line to find the y value of G

y= -2(3/5) + 2 =  -6/5 + 10/5 =  4/5

G = ( 3/5 , 4/5)

DG =  sqrt  [ (3/5)^2  + (4/5)^2 ]  = sqrt  [9/25 + 16/25 '  =  sqrt  [25/25]  = sqrt [ 1 ]  =  1

(90" * 75") / (24" * 12") = 23.4375 tiles

LOL!!!