All Answers 356032 Answers

Your logic makes perfect sense to me. :))

I think that if 10^(n+1) + 1 contains at least 1 perfect square factor, then this statement can be disproven. 

=^._.^=

Thanks Geno,

I think your answer is cool too :)

Attn:  ibhar,

Thank you for responsing to Gino's answer so nicely.   

BUT

You need to think.

Your statement:   "I also assume that some non-odd primes work"

1) Firstly, tell me some non-odd primes.

2) second show me with one of them what the average of its factors is. 

3) discuss your answers.

YES  you are expected to answer me :)

How many 4-digit numbers have the second digit even and the fourth digit is at most  twice the second digit? (Here, digits are read from the left, so the first digit is the leftmost digit, and so on.)

0          0                     9*10 = 90

2,      (0,1,2)               9*10*3 = 270

4,       (0-8)                 9*10*9 = 810

6       (0-9)                  9*10*10=900

8        (0-9)                 9*10*10=900

90+270+810+1800 = 2970

Thanks, :)

Unfortunately I do not think this question will get my attention.

It looks very time consuming and it doesn't look like my kind of question. 

I hope someone else can help you here. :)

All counter examples will be of the form  

a^2b=bc^2    

as I have described above.

There is most likely an infinite number of them.

YAY CPhill!

Oh! That's really cool! I haven't thought of the odd primes.

I also assume that some non-odd primes work and as you say, some even numbers work too. Is there a specific way of describing them or is it just at random?

Thank you so much,geno3141!

Ah... I forgot that you roll ONLY 1 dice. Sorry... 🤦

I agree with EP on this problem; there is no way to paint 2 and 2 because

there is only one 2.

Those that work:

   1,2     1,3     1,4     1,5     1,6

   2,3     2,4     2,5     2,6

   3,4     3,5     3,6

   4,5     4,6

   5,6

The one that doesn't:  1,4

THAT is a REALLY high interest rate !           This is  16  + 3 + 1/3 + 1/6   = 19.5   'quarters'        interest per quarter = .445 / 4

80 000 ( 1 + .445/4)^(19.5)

You forgot 2 and 2 EP, thus, there are 13 working cases out of 15 total cases. 

you cannot paint the 1 and the 4  as the two faces

there are    6 c 2   ways to paint two faces = 15 ways

    and ONE way (1 and 4) that you cannot paint    = 14 ways

GCF of   840   and 1008  is   168  stamps per page

Put in x = -5 and solve for y

-3(-5) + 7y = 5 (-5) + 2y

40 = -5y

y = - 8

Every odd prime number has two factors and only two factors, 1 and the prime number.

The result of adding 1 to the odd prime number is an even number; dividing this 

answer by 2 results in an integer.

Now, you have an infinite number of answers.

And yet, there are more; some even numbers also fulfill the requirement ...

I wonder if    10    would count as one of the possibles ?

Yah.... I believe you are correct....   the wording was a bit clumsy, so I included the equation to calulate ( as you did) ...

    Just put in 15000 for PV    and then calculate  'C'

15 , 29 , 51 , 92 , Total =187  

I agree with EP!  If repetitions are allowed, I calculate: 3,150 such 4-digit integers.

Hi EP: I think this is an ordinary annuity, with a PV of $15,000, from which you would receive quarterly payments @ 5% comp. quarterly for 10 x 4 ==40 quarters. If that is right, then the quarterly payments I calculate come to ==$478.82 for 10 years, or 40 quarters. However, the question is vey vague!

Have responded. As I said, I will work on characterizing all numbers that serve as counter-examples.

I'm curious to know an approach to this post too.

jsaddern

Thank you! I profoundly appreciate your response.

I understand your logic and see the four counter-examples you provide. I will continue to work on describing all numbers that work as counter examples. Thanks Melody,

jsaddern

Write as system: 

x = cost of first hotel

y = cost of second hotel

0.075x + 0.095y = 542.25

y = x - 1000

SOlving, we find \(\color{brown}\boxed{x=3,750, y = 2,750}\)

This, I believe is an ordinary annuity

PV = C * [  1 - (1+i)^-n ] / i

PV = present value        C = 15 000       i = interest per period = .05/4         n = periods = 4 x 10 = 40

pluggiing in these numbers shows PV = ~  $ 469904