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Here's how I view this....

5^1 / 8 = (8 - 3)^1 / 8   

Every term in the expansion is a multiple of 8 except the last  which is

(-3)^1 / 8  = (-3)^1  mod 8  = 5

Similarly

5^2 /8  =  (8 - 3)^2 / 8

Again we only need consider the last term which is

(3)^2/8  =  (3)^2 mod 8  =   1

etc....

So...we only need to consider  the last term in each subsequent power expansion and note that

(3)^n mod 8 =  1      and

(-3)^n mod 8 =  5

So

5^137 / 8    =  (8 - 3)^137 / 8      will  have (-3)^137 / 8  as a last term

And the remainder is  (-3)^137 mod 8  =   5

Sorry, it's wrong, b means base b, not multiply by b

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Wow! I wasn't active in such a long time, I'm so sorry. Finally, I saw this post! Happy birthday CPhill, hope you have a blast!

Enjoy my recipe:

Oreo flavor!

Good job, tertre!!!!!

Melody, you're on the right track, but here is my attempt:

You're right about the odd and even switching,

5^3=125

5^4=625

5^5=last three digits of 125

5^6=last three digits are 625

So, for odd powers of five, the last three digits are 125 and for even powers five, the last three digits are 625. By the divisibility rule for 8, and we have \(\frac{125}{8}\) since 137 is odd. Thus, the remainder is \(\boxed{5}.\)

Here's my best attempt.....

I believe that the answer is 1

When two circles are  tangent....a line drawn  through their centers will pass through the point of tangency...so....let the center of the the circle be (x,y)  and the radius = r

The distance for the center of the circle to  (0,5) will just = r

The distance from the center of this circle to the centers of the given circles will be r + 1

We have this system

(x - 4)^2 + y^2 =  (r + 1)^2

(x + 4)^2 + y^2 = (r + 1)^2

x^2 + (y - 5)^2 = r^2

This is a little messy to solve  [ but not impossible]...so I used WolframAlpha to generate the solution

The center of the circle is (x, y) =   (0, 5/3)    and r = 10/3

Here's a graph :

PS....I can show you how to solve the equations if need be...but it's a little lengthy!!!!

1    5,   

2    25,

3    125,           odd

4    625,             even

5    3125,           odd

6    15625           even

7     78125             odd

8    390625           even

ok I have a pattern

5^137 = 1000n+ 125 = 125*8*n  +   17*8     +  1  =  8(1000n+17) +1       n has a specific integer value

Yes I made a small stupid numeric error that i am quite sure hatchet288 was smart enough to correct.

This last line should have been

5^137 = 1000n+ 125 = 125*8*n  +   15*8     + 5  =  8(1000n+15) +5       n has a specific integer value

so

\(5^{137}\div 8 \)        has a remainder of 5

It was no big deal, any question asker should have been able to pick up that little error for themsleves.

BUT Good effort Tertre, it was a good catch.

We need to find the slope of the line

dy/dx =   dy/dt  /  dx/dt  =  cost / - sin t  =   - cot t

At  t = pi/4, the slope is    - cot (pi/4)  =  - 1

The normal line will have a slope of 1

The point is given by   ( cos pi/4, sin pi/4) =  (  sqrt (2)/2, sqrt (2)/2 )

So....the equation of the tangent lne is

y = 1 (x - sqrt (2) /2 )  +  sqrt(2)/2 

y =  x

Here's the  graph :

Here's my best attempt, Cupcake.......someone else should verify this....

Expected value =

Prob of red * payoff + Prob of green * payoff + Prob of blue * payoff + Prob of yellow * payoff  =

2/12 ( 2)  + 2/12 ( 2) + 2/12 ( 2) +  6/12 (-3)  =   - 1/2

"D"

I believe that....to be a fair game, the payoff for the  yellow should only be losing 2 tokens...

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Yes,  THAT is what we wanted...... the value (present of future) of his total paychecks for 5 years starting at 64000 per year and increasing 8 % per year.     405 K is incorrect.  (you gave him an extra 8%  )

Another point: An increasing/decreasing annuity is an annuity to which either additional payments or an additional percentage(such as the inflation rate) is added to the regular amount or to the regular percentage.

  That is why I set r=0  and g = .08

... it is equal to

64000 + 64000(1.08) + 64000(1.08)^2 + 64000(1.08)^3 + 64000(1.08)^4

64000 (1 + 1.08 + 1.08^2 + 1.08^3 + 1.08^4) = $ 375462.46

C(50, 6) = 15890700 =  the number of winning "sets" of numbers

Only one will win.....so [ assuming one winner]......the probability of winning is   just 1/15890700

And the probability of losing is  [ 1 - prob of  winning ]

Expected value =

Prize money * probability of winning - Cost to play * probability of  losing =

2,000,000 * (1 / 15890700)   -   2  * [ 1 -   1/15890700]   ≈ -1.87  =   -$1.87

"A store sells cookies at 8 cookies for $2.00. If it changes the price to 10 cookies for $3.00, what is the percent increase in price per cookie?"

2.00 / 8 =  .25 per cookie

3.00 /10  = .30 per cookie

Price increase is

[ 30 - 25 ] / 25  x  100%  =

[ 5 ] / 25    x 100% =

1/5 x 100% =

[100 / 5] %  =

20 %

225 km / 45 km/hr = 225/45 = 5 hours   + 15 minutes of stop = 5 hr 15 min  

1:10 PM + 5 hr 15 min = 6:25 PM   or   18:25     

 876-720 = 156 degrees

156 x pi/180 = 13/15  pi   

         for a  reference angle    2/15  pi    Radians

Just Multiply: $321.50*1.5=$482.25

\(f^{-1}(g(x)) = x^3-1\\ g(x) = f(x^3-1)\\ x = g^{-1}(f(x^3-1))\\ 7 = 2^3-1\\ g^{-1}(f(7)) = g^{-1}(f(2^3-1)) = 2\)

\(A=32x^2+8x = \ell \cdot w\\ A = 8x(4x+1)\\ \text{Given the info you've provided there's really no further we can go}\\ \ell = 8x \\ w = (4x+1)\ \text{all you can really say about }x \text{ is }\\ x>0 \text{ since length must be positive}\)

Let me take a crack at this.

First, bring the one down.

Then, -3*1=-3, so it goes underneath the -13.

Add, -13+(-3)=-16

So, -3*-16=48

Finally, -48+48=0

x-16, (A). I don't know if I'm correct, though.

Yep, it is for you, CPhill! Have a fun time!

Happy birthday, CPhill! Enjoy!

Two tied 8th notes -  translation and reflection

Two half notes - translation and reflection

The scale factor of the smaller box to the larger   =   ∛ [ 27 /64]  =  3/4

The surface area  of the smaller box =

(surface area of larger box) * (scale factor)^2  =

(160) * (3/4)^2  =

(160) * 9/16  =  

90 in^2

EP: What you calculated is the FV of an "ordinary annuity" at the END of the period. What I calculated is the also the FV of an "annuity due", or at the BEGINNING of the period!!. In other words, my figure is your figure x 1.08 =$375,462.46 x 1.08 =$405,499.46. By the way, the PV of these 5 payments =$255,533.44.

Another point: An increasing/decreasing annuity is an annuity to which either additional payments or an additional percentage(such as the inflation rate) is added to the regular amount or to the regular percentage.

Here is an example: Regular payment=$1,000. Additional payment=$100. Rate=8%. Period =40 years.

OR: Regular payment =$1,000. Rate=8%. Inflation rate=4%. Period =40 years. 

Regular annuity would NOT have an additional $100 or inflation rate of 4%.

Area = side^2

So

64 = side^2     take the square root of both sides

8 (ft) =  side

8 ft =  8*12  =  96 inches

If each side was 1 ft shorter   =   7 ft on a side