Melody

Thanks Heureka,

I understand about half of that,  at least I now know what 9 being its own inverse means.

So 

9*9=1 (mod m)

81=1 (mod m)

so

80=0 (mod m)

Factors of 80 are     1, 2,4,5,8,10,16,20,40,80

This won't makes sense for 1,2,4,5, or 8

9*9 =81 = 1 (mod 10)

9*9 =81 = 1 (mod 16)

9*9 =81 = 1 (mod 20)

9*9 =81 = 1 (mod 40)

9*9 =81 = 1 (mod 80)

now 3*3=9 so this will stay 9 for any of the above mods.

So the possible values of m are  10,16,20,40, and 80       There are 5 of them

Is this right Heureka or am I not understanding something?

1)   The only one that is 10% is the last one   10% of 40 is 4

2) I'd go for the last one there too.

Heureka,  could you please tell me what 

"9 is its own inverse" means 

I do not know why you did not see CPhill's answer as he was answering your post and the new answer would have been displayed in your watchlist. He did answer you before you posted your answers

But it was not much before so maybe you had not seen it yet.

Anyway, I'll accept what you say at face value and I appologise.

 In the future please follow the guidelines that I have written:

1) Reposts should take the answerer back to the original question (with a link) , so that it does not get answered twice without the answerer's knowledge.

2) One question per post. Typed in whenever possible.

3) Learn from it, ask questions if you do not understand.

4) Say thanks and give points.

5) Then after you have learned all you can from your first answer, you can post a new question.

Thanks.

Yes the problem is with the askers not the answerers. I know the answerers are all very generous people who just want to help facilitate learning. All I have suggested is that you think first, answer second :)

Thanks x^2,

You do not need to take down your original post, you didn't say anything bad :)

Gavin I think I follow what you are saying and it sounds logical but a cone IS a pyramid.

It is a circular pyramid. So when you say a pyramid I think you are refering to a square or a triangular pyramid..?

\(\displaystyle \lim_{x\rightarrow ∞} \sqrt{\frac{16x}{4x+9}}\\ =\displaystyle \lim_{x\rightarrow ∞} \sqrt{\frac{16x\div x}{(4x+9)\div x}}\\ =\displaystyle \lim_{x\rightarrow ∞} \sqrt{\frac{16}{4+\frac{9}{x}}}\\ = \sqrt{\frac{16}{4+0}}\\ =\sqrt4\\ =2 \)

Hi X-squared,

My observations was not aimed at anyone in particular and certainly not at you. You have been providing great answers for a long time and I am very grateful to you.  I do not contribute as much as I used to but I continue to be passionate about this forum and I know that some question askers learn a great deal from the fabulous answers that you and others give.  It is these students that inspire us and make this whole site worthwhile. 

CPhill has told me that if he teaches even just one student and helps to foster in them a great love of mathematics that that is all the reward he needs.  (I hope you do not mind me paraphrasing Chris)

X-squared this is nothing wrong with you being pleased that your own score is increasing. I like seeing mine increase too. It is all a part of the 'game'. Do not criticize yourself for it. :)  Also I do not intend to be critical of people who pick questions that they personally are interested in. There is nothing wrong with that either. 

I just wanted answerers to think a little more before they answer. That is all.

For my own part I do not intend to answer questions if they are posted in multiples. But sometimes the question will interest me and I will answer anyway.  I will also make more of an attempt to only half answer questions but like many of you, it is always very tempting to give whole answers, we are precise people, we like to finish what we start. 

Plus sometimes I get frustrated because I give a half answer for a very good reason, and then someone else comes in and completes it. I do not mind if it is a child who is learning but it does frustrate me when it is someone with higher abilities.

I'd really like it if we could encourage the question askers to interact with us more  It is great when question askers challenge  or question my answers. It shows me that someone at the other end is really trying to learn!

I expect all you wonderful answerers feel the same way.

I will finish now with a huge thank you to all the wonderful people who put so much effort into teaching people here. And also a big thanks to those students who seriously learn from us and make our time here all worth while.

I normally will not answer questions when there are several asked on the same post but I am making an exception here because a much respected member has asked me too.  (He wants to see how it can be answered for himself)

From a point interior to a regular hexagon, perpendiculars drawn to the sides on the hexagon have lengths, in order of size of 3, 6, 7.5, 13.5, 15 and 18. How long is the side of this hexagon?

A hexagon has got 6 sides. Opposite sides are parallel. So the perpendicular from a point to one side will also be the perpendicular from the point to the opposite sides.

If you look at the given lengths you can see that 

3+18=21

6+15=21

7.5+13.5=21

So 21 is the perpendicular distance between the parallel sides.

So the centre of the hexagon is 10.5 perpendicular units from each side.

The angle sum of a hexagon is 180*(6-2) = 720 degrees

Each internal angle is 720 / 6 = 120 degrees.

Half of each internal angle is 60 degrees.

So this is what we have.

\(tan60^\circ=\frac{10.5}{x}\\ \sqrt3=\frac{10.5}{x}\\ x=\frac{10.5}{\sqrt3}\\ x=\frac{21}{2\sqrt3}\\ x=\frac{21\sqrt3}{2*3}\\ x=\frac{7\sqrt3}{2}\\ \text {the side length is 2 * x}\\ Side\;length\;=7\sqrt3\;units \\ Side\;length\;\approx 12.124\;units \\ \)

Here is the pic with the original figures. This pic was not needed to answer this question but it may help you to understand better.