Portfolio Type 1 Investigating Divisibility HELP!

[eel7x6]

I am doing my Maths portfolio on investigating divisibility atm. I am really stuck on question 3 and 4.

These are the portfolio questions

1. Factorize the expression P(n)=n^x-n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k+1)-P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x.

2. Explain how to obtain the entries in Pascal's triangle...State the relationship between the expression P(k+1)-P(k) and Pascal's traingle. Reconsider your conjecture.

Write an expression for the xth row of the Pascal's Triangle. You will have noticed that (x r) = k, k is a natural number. Determine when k is a multiple of x.

3. Make conclusions regarding the last result in part 2 and the form of proof by inductiton used in this assignment. Refine your conjecture if neccessary, and prove it.

4. State the converse of your conjecture. Describe how you woul prove whether or not the converse holds.

The ones in bold are the questions I dont get, especially the 3rd and 4th question. (what is converse?) The italic part - I do not know how to use words to describe it...

Plz help it is due this wednesday!!!

[Scade]

I've done the same IA about a month ago, but don't know how went it well. Anyway I'll try to provide some hints Writing the expression for the xth row is just basically writing the expression you would use to get any row in pascal's triangle (provided you're using the n C r method). In the next part k stands for just a standard coefficient. If you've done the previous parts correctly it should be pretty obvious when k is a multiple of x.

Again if you've done everything correctly up to part 3 the conclusions you make should be pretty obvious (it relates directly to your conjecture). I wasn't quite sure what they meant with the part of making conclusions about the form of proof by induction. Anyway I talked about the restrictions it places, when it is generally used and that kind of stuff. Refining your conjecture should be pretty obvious and proving it is also rather easy.

You should check converse from some dictionary. Basically it is on accordance with your conjecture, and just derived from it.

As a general advice you might want to do some search in the internet. At least it helped me quite a lot. Remember just not to copy anything, only see what information you find and then think how you can apply it.

[eel7x6]

for the second part of the 2nd question, I said k is a multiple of x when x is a prime number. is that right?

[Scade]

Yup, that was my statement also. In the end if you do the questions in order it should sort of open up by itself.

[diabolicalangle]

nmv, im being a dunce

Edited April 5, 2008 by diabolicalangle

[Scade]

i need help on this topic tooo

Please ask specifically what kind of help you need. I don't think anyone will be able to provide help when it is not specified where you need help

[appleme]

4. State the converse of your conjecture. Describe how you woul prove whether or not the converse holds.

The ones in bold are the questions I dont get, especially the 3rd and 4th question. (what is converse?)

I really need help with this bit too... a converse is the opposite - so i think that if your conjecture is: all even numbers have the letter e in them (silly example i know), the converse would be numbers that have the letter e are even... can anybody verify this?

and when it says describe how you would prove it instead of prove it i really dont know what they are asking for! do you say something like i would try each number? because that just sounds silly

Edited May 9, 2008 by appleme

[Amuh]

please some one post sumfn more on this topic...please please i also need help...i am stuck with the first point...

[appleme]

please some one post sumfn more on this topic...please please i also need help...i am stuck with the first point...

the first one?

its basically looking at each n^2-n, n^3-n etc. and proving it works by induction...

[Amuh]

can some one here send me a copy of what they did..i just want to get a rough idea..coz i am duffer in math HL

[Camlon]

Do any of you know how to prove the converse? I'm able to prove 3, but the converse is much harder.

Edited May 19, 2008 by Camlon

[Scade]

If you look at the wording, it doesn't say that you need to prove it, just state how you would do it. My hint is to not use induction, but contradiction.

[Camlon]

Yeah, but shouldn't I at least tell how to do it generally, or could I just write.

To prove that Q=p(k 1))- p(k) is not divisible by x, use proof by contridiction and assume that evey term of Q is divisible and then end up with a contridiction. I didn't really use induction for 3 either, I just proved it and found a way to make it a proof with induction.

Edited May 19, 2008 by Camlon

[Camlon]

I showed that it was not true for x=561, using Little Theorem of Fermat. Everybody was expecting it to be true, but it's not. As I'm the only further maths student at my school I think I'm the only one who got that.

[pdenny29]

I showed that it was not true for x=561, using Little Theorem of Fermat. Everybody was expecting it to be true, but it's not. As I'm the only further maths student at my school I think I'm the only one who got that.

561 actually isn't prime and wouldn't work.

[shrekmak]

no, this guy has a point.

561 is a counter example for the converse of your conjecture.

do some research.... omg, he just saved me 10pts.

[nba favor]

I've done the same IA about a month ago, but don't know how went it well. Anyway I'll try to provide some hints Writing the expression for the xth row is just basically writing the expression you would use to get any row in pascal's triangle (provided you're using the n C r method). In the next part k stands for just a standard coefficient. If you've done the previous parts correctly it should be pretty obvious when k is a multiple of x.

Again if you've done everything correctly up to part 3 the conclusions you make should be pretty obvious (it relates directly to your conjecture). I wasn't quite sure what they meant with the part of making conclusions about the form of proof by induction. Anyway I talked about the restrictions it places, when it is generally used and that kind of stuff. Refining your conjecture should be pretty obvious and proving it is also rather easy.

You should check converse from some dictionary. Basically it is on accordance with your conjecture, and just derived from it.

As a general advice you might want to do some search in the internet. At least it helped me quite a lot. Remember just not to copy anything, only see what information you find and then think how you can apply it.

What is actually the relationship between the expression P(k+1)-P(k) and Pascal's Triangle?

[AlphaMagnum]

I'm having a bit of trouble with the second two questions on this one, but I'll put up the full set before I get to that.

1. Factorize the expression P(n) = n^(x) - n for x {2,3,4,5}. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your results by showing whether P(k+1) - P(k) is always divisible by x. Using appropriate technology, explore more cases, summarize your results, and make a conjecture for when n^(x) is divisible by x.
   
2. Explain how to obtain the entries in Pascal's Triangle, and using appropriate technology, generate the first 15 rows. State the relationship between the expression P(k+1) - P(k) and Pascal's Triangle. Reconsider your conjecture and revise if necessary. Write an expression for the xth row of Pascal's Triangle. You will have noticed that nCr(x,r) = k, k is a natural number. Determine when k is a multiple of x.
   
3. Make conclusions regarding the last result in Part 2 and the form of proof by induction used in this assignment. Refine your conjecture if necessary, and prove it.
   
4. State the converse of your conjecture. Describe how you would prove whether or not the converse holds.
   

I've gotten through numbers 1 and 2, and my two problems with those were:

• How do I appropriately use technology? Do you have any suggestions for how best to show that I used it?
  
• Starting from #1, I said that the expression is divisible when x is prime, so do I just say that the Pascal's Triangle results are in accordance with that conjecture?
  

I'm stuck on numbers 3 and 4 entirely, because I'm not entirely sure as to what they're asking.

For 3:

• I can talk about proof by induction and why it works here but not necessarily elsewhere.
  
• I'm not sure about what kind of conclusions they want me to make.
  
• I'm still looking at how I'd like to prove it, but I'm not sure if I should use a proof by contradiction (need to figure out how first) or some other type.
  

For 4:

This question doesn't actually ask for much, so I'm thinking that I should be fine with it once I determine a neat statement of my conjecture from which I can find the converse.

Any help with this would be really appreciated,

AlphaMagnum

[pumpkinora]

I am doing my Maths portfolio on investigating divisibility atm. I am really stuck on question 3 and 4.

These are the portfolio questions

1. Factorize the expression P(n)=n^x-n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k+1)-P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x.

2. Explain how to obtain the entries in Pascal's triangle...State the relationship between the expression P(k+1)-P(k) and Pascal's traingle. Reconsider your conjecture.

Write an expression for the xth row of the Pascal's Triangle. You will have noticed that (x r) = k, k is a natural number. Determine when k is a multiple of x.

3. Make conclusions regarding the last result in part 2 and the form of proof by inductiton used in this assignment. Refine your conjecture if neccessary, and prove it.

4. State the converse of your conjecture. Describe how you woul prove whether or not the converse holds.

The ones in bold are the questions I dont get, especially the 3rd and 4th question. (what is converse?) The italic part - I do not know how to use words to describe it...

Plz help it is due this wednesday!!!

[pumpkinora]

I'm having a bit of trouble with the second two questions on this one, but I'll put up the full set before I get to that.

1. Factorize the expression P(n) = n^(x) - n for x {2,3,4,5}. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your results by showing whether P(k+1) - P(k) is always divisible by x. Using appropriate technology, explore more cases, summarize your results, and make a conjecture for when n^(x) is divisible by x.
   
2. Explain how to obtain the entries in Pascal's Triangle, and using appropriate technology, generate the first 15 rows. State the relationship between the expression P(k+1) - P(k) and Pascal's Triangle. Reconsider your conjecture and revise if necessary. Write an expression for the xth row of Pascal's Triangle. You will have noticed that nCr(x,r) = k, k is a natural number. Determine when k is a multiple of x.
   
3. Make conclusions regarding the last result in Part 2 and the form of proof by induction used in this assignment. Refine your conjecture if necessary, and prove it.
   
4. State the converse of your conjecture. Describe how you would prove whether or not the converse holds.
   

I've gotten through numbers 1 and 2, and my two problems with those were:

• How do I appropriately use technology? Do you have any suggestions for how best to show that I used it?
  
• Starting from #1, I said that the expression is divisible when x is prime, so do I just say that the Pascal's Triangle results are in accordance with that conjecture?
  

I'm stuck on numbers 3 and 4 entirely, because I'm not entirely sure as to what they're asking.

For 3:

• I can talk about proof by induction and why it works here but not necessarily elsewhere.
  
• I'm not sure about what kind of conclusions they want me to make.
  
• I'm still looking at how I'd like to prove it, but I'm not sure if I should use a proof by contradiction (need to figure out how first) or some other type.
  

For 4:

This question doesn't actually ask for much, so I'm thinking that I should be fine with it once I determine a neat statement of my conjecture from which I can find the converse.

Any help with this would be really appreciated,

AlphaMagnum

Hi guys.

I too am stuck and am not sure where to start on number one. Can anyone point me in the right direction...not looking for the answer, just some guidance. Mine's due in tomorrow.

HELP