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#1 2007-03-05 16:39:49

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Jane’s exercises

Two integers are said to be relatively prime iff their highest common factor is 1. Thus, 2 and 5 are relatively prime, as are 4 and 9.

A result called Bézout’s identity states that if a and b are nonzero integers that are relatively prime, there exist integers x and y such that ax + by = 1. {Note that x and y can be negative as well as positive.)

1. Consider the number 60. 60 is divisible by 4 and 5, and 60 is also divisible by 4×5 = 20. However, 60 is divisible by 4 and 10 but not by 4×10 = 40. 4 and 5 are relatively prime, whereas 4 and 10 are not.

Using Bézout’s identity (or otherwise) prove that if an integer c is divisible by both a and b, where a and b are nonzero, relatively prime integers, then c is also divisible by the product ab.

2. Prove that the product two consecutive even numbers is divisible by 8.

3. Hence (or otherwise) prove that if n is an odd integer that is not divisible by 3, n[sup]2[/sup]−1 is divisible by 24.
­


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#2 2007-03-05 16:45:34

ganesh
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Registered: 2005-06-28
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Re: Jane’s exercises

The one easiest to answer is #2, since any two consecutive even numbers are such that one of them is certainly divisible by 2. Hence, the product would be divisible by 8.

Nice questions, JaneFairfax, I suggest you name the topics based on the subjects posted and suffix or prefix them Jane-1 etc. depending on the number of the post!

Your posting in the Exercises section is much appreciated!!!


Character is who you are when no one is looking.

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#3 2007-03-05 17:09:02

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

Well, you’re on the right track with your answer to #2, but you need to elaborate just a little bit.

I’m intending to use this thread to post some interesting exercise questions on various math topics (and various math levels); some of the questions will be interrelated. Thus I’ve posted my first three exercise questions. smile


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#4 2007-03-05 17:37:10

pi man
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Registered: 2006-07-06
Posts: 251

Re: Jane’s exercises

An even number can be expressed as 2x.   The next even number would be 2x + 2.   The product would be 4x^2 + 4x = 4x( x+ 1).   Either the x or the (x+1) has to be even and therefore divisible by 2.  Combine that with the  4, and you have your 8.

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#5 2007-03-06 03:58:29

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

Correct. up


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#6 2007-03-06 04:23:52

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Jane’s exercises

n² -1 can be factorised into (n+1)(n-1). We are given that n is odd, and so n+1 and n-1 are both even. Hence, by the answer to 2), n² -1 is divisible by 8.

We are also told that n is not divisible by 3. This means that it takes either the form 3k+1 or 3k+2, where k is an integer.

For the case of 3k+1, this would mean that n-1 was equal to 3k and so divisible by 3.
For the case of 3k+2, this would mean that n+1 was equal to 3k+3 and so divisble by 3.

Therefore, one of (n+1) and (n-1) is always divisible by 3 and so n²-1 is always divisible by 3.

We have previously shown that n²-1 was also divisible by 8, and combining these pieces of information shows that n²-1 is divisble by 24.


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#7 2007-03-06 08:39:16

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

Excellent job. up

Now someone try #1? It’s not that difficult. roll


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#8 2007-03-10 18:13:00

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

More exercises from me. big_smile

4. If x is real, what is 0[sup]x[/sup]?

5. If x is a real number, [x] denotes the greatest integer less than or equal x. Prove that for any real numbers x and y,

In other words, the greatest-integer function is a superadditive function. (Hint: If n is any integer such that nx, then n ≤ [x].)

6. A cycloid is the curve traced by a fixed point on a circle rolling along a straight line. If the radius of the generating circle is r, the parametric equations of the cycloid are

Find the area bounded by one arch of the generated curve (0 ≤ θ ≤ 2π) and the x-axis.

Last edited by JaneFairfax (2007-03-10 18:18:22)


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#9 2007-03-19 05:11:53

Kurre
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Registered: 2006-07-18
Posts: 280

Re: Jane’s exercises

1. every number that isnt a prime can be factorized in its prime factors, for example 36 can be factorized in 2*2*3*3.
so the number c can be factorized in the prime factors P1*P2*P3.....Pn
a and b are relatively prime so they cant have one same prime factor, but since c is divisible by both a and b, they must be created by one or some of the prime factors in c. therefor a*b can never be more than c.

for example a can be P1*P2 and b can be P3*P4. if we divide c with a and b there will still be P5*P6*....*Pn left.
but if a is P1*P2, b cant be for example P2*P3 since then both will have P2 as a prime factor.

would this be a correct proof?? still practising smile

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#10 2007-03-19 12:12:54

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

You’re more or less on the correct line of thought. However the problem is to show that c is divisible by ab. It’s true that ab is less than or equal to c, but this doesn’t show that c is divisible by ab.

I’ve already given a short solution using Bézout’s identity. smile


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#11 2007-03-19 13:45:04

JaneFairfax
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Registered: 2007-02-23
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Re: Jane’s exercises

Last edited by JaneFairfax (2007-03-19 19:19:07)


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#12 2007-03-19 13:56:22

mathsyperson
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Re: Jane’s exercises

I thought that 0[sup]0[/sup] was indeterminate? There's a problem because 0^n = 0, but n^0 = 1, so there we have a contradiction.


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#13 2007-03-19 19:18:33

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

0[sup]0[/sup] is defined as 1. Try it on a calculator. roll

And

so the graph is not right-continuous at x = 0.

Last edited by JaneFairfax (2007-03-19 19:22:26)


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#14 2007-03-19 22:09:12

mathsyperson
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Registered: 2005-06-22
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Re: Jane’s exercises

What calculator are you using? Any that I try just tell me Ma Error or something similar.


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#15 2007-03-19 23:18:26

Kurre
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Registered: 2006-07-18
Posts: 280

Re: Jane’s exercises

JaneFairfax wrote:

You’re more or less on the correct line of thought. However the problem is to show that c is divisible by ab. It’s true that ab is less than or equal to c, but this doesn’t show that c is divisible by ab.

I’ve already given a short solution using Bézout’s identity. smile

well since ab is created by the prime factors in c, c must be divisible by ab?

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#16 2007-03-20 13:49:50

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

mathsyperson wrote:

What calculator are you using?

A decent one. tongue

Kurre wrote:

well since ab is created by the prime factors in c, c must be divisible by ab?

I’ve read your proof again and I can see what you’re trying to get at – so, yes. smile

Proofs involving writing out long sequences of primes can sometimes become blurred with details and hard to follow – I generally avoid them if I can find alternative proofs. wink


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#17 2007-03-20 15:26:23

Zhylliolom
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Registered: 2005-09-05
Posts: 412

Re: Jane’s exercises

Letting 0[sup]0[/sup] be 1 is a standard convention, which helps eliminate special cases where some theorems break down. There are some combinatorial and set-theoretic justifications for this convention. However, in analysis 0[sup]0[/sup] is treated as an indeterminate. Moreover, consider x[sup]x[/sup] = e[sup]x ln x[/sup]. There is no real number corresponding to ln 0, so x[sup]x[/sup] is not defined at x = 0; in other words 0[sup]0[/sup] is undefined. The calculator says it is equal to 1 because of the common convention mentioned above; but a real decent calculator would notify you something like "Warning: 0^0 replaced by 1" wink.

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#18 2007-04-03 03:12:37

Kurre
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Registered: 2006-07-18
Posts: 280

Re: Jane’s exercises

Thanks for the tip Jane.
I really liked exercise 2 and 3, i really need to practise proving these types of problems. i would really appreciate if you could create more exercises like them tongue

Last edited by Kurre (2007-04-03 03:12:56)

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#19 2007-04-03 04:22:38

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

Well, I’ve been trying to create some more exercises for this thread. Here is something I just this moment made up. big_smile

7. Let f(x) be a periodic odd function with period n. In other words, f(x) satisfies f(x) = −f(x) (odd function) and f(x+n) = f(x) (periodic with period n). If n is an odd integer greater than 1, prove that

8. Use the above result to show that

NB: For #8 you must use the result in #7. wink

Last edited by JaneFairfax (2007-04-16 10:27:46)


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#20 2007-04-04 11:12:50

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Jane’s exercises

right?


Numbers are the essence of the Universe

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#21 2007-04-04 11:15:36

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Jane’s exercises


Numbers are the essence of the Universe

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#22 2007-04-04 20:13:19

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

Stanley_Marsh wrote:

right?

No. You have the right idea, though. smile

Stanley_Marsh wrote:

True, but as I’ve stated, I want you to use the result in #7 to do #8. big_smile


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A: Click here for answer.

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#23 2007-04-05 02:29:10

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: Jane’s exercises



Numbers are the essence of the Universe

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#24 2007-04-09 00:17:12

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

That’s the way to do it. up


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#25 2007-04-11 17:08:01

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Jane’s exercises

9. Prove that a monotone-increasing sequence of real numbers that is bounded above converges to its least upper bound.

10. Give an example of a monotone-increasing sequence of rational numbers that is bounded above which does not converge to a rational number.

11. Show that nevertheless a monotone-increasing sequence of rational numbers that is bounded above is a Cauchy sequence.


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