[Maths boy]

hey, im just wondering if anyone can help me with this 1st year maths problem in the number and structure unit.

a) Using the Euclidean algorithm find the highest common factor of 1905623 and 2766853. Find also two integers h and k such that 1905623h + 2766853k = the HCF you have found.

b) Invent a linear congruence problem of the form with your own choice of c where c >1500. Find two different solutions to your problem i.e. find two different pairs of values for x and y.

c) Invent a different linear congruence problem with 1905623 and 2766853 which does not have any solutions (i.e. with a different value of c.) Explain why it does not have any solutions.

[aznkid66]

Lol, discrete mathematics is one of the option topics for IB Mathematics HL.
I haven't formally learned the procedure yet, but I think I understand it being that it's an algorithm. I think I can help you with the algorithm, but I'm not sure if that's what you need help on since you haven't explained what you got up to in your post ^^; Plus, being that I haven't been formally taught it, I might make up some words...

Basically, the Euclidean algorithm is the way to find the "HCF", or greatest common factor, of two integers. For example, the HCF(6,4)=2.

http://en.wikipedia.org/wiki/Euclide...ithm#Procedure

"The Euclidean algorithm is iterative, meaning that the answer is found in a series of steps; the output of each step is used as an input for the next step."

Label the larger numer 'a' and the smaller number 'b'.
To find HCF(a,b), write the following equations:
a = kb + c

Spoiler:

Show

where kb<a and (k+1)b>a. Basically, you're dividing 'a' by 'b', where 'k' is the integer solution and 'c' is the remainder.

Find HCF(b,c)

Spoiler:

Show

so plug in 'b' and 'c' that you obtained in the previous equation into where 'a' and 'b' were. You get a new equation, and by extension new values for the next 'k' and the next 'c'.

When r=0, stop. The previous 'r' is the HCF of the two numbers you started with.

Examples:

Spoiler:

Show

HCF(246,48):

246 = 5*48 + 6
48 = 8*6 + 0

HCF(246,48)=6

Spoiler:

Show

HCF(499,37):

499 = 13*37 + 18
37 = 2*18 + 1
18 = 18*1 + 0

HCF(499,37)=1 (they are coprime)

As you can see, it can be much faster than prime factorization and repeated division for large numbers with large factors :P

I hope this helps you with the first part of part (a), if you were stuck there. The rest of part (a) is just the extended Euclidean algorithm. If you have any more trouble, just ask!

[aznkid66]

(Original post by Maths boy)
Thank you, can u help me with this question aswell

Considering the set
a) Prove that S forms a ring.

b) If two members of this ring multiply to make an integer which is a prime number, what form will the factors take?

c) Find two numbers which are prime in the integers but which are composite in this ring and factorise them in this ring.

d) Find two distinct members of this ring which multiply together to make 1.

e) Find as many units as you can in this ring. Explain how you know that they are units.

Huh, it looks like the set that was mentioned did not appear. Could you type up or attach an image of the set S?

[Maths boy]

sorry:
considering the set S={a+broot8: a,b are integars}
a) Prove that S forms a ring.

b) If two members of this ring multiply to make an integer which is a prime number, what form will the factors take?

c) Find two numbers which are prime in the integers but which are composite in this ring and factorise them in this ring.

d) Find two distinct members of this ring which multiply together to make 1.

e) Find as many units as you can in this ring. Explain how you know that they are units.

[Maths boy]

(Original post by aznkid66)
Huh, it looks like the set that was mentioned did not appear. Could you type up or attach an image of the set S?

sorry:
considering the set S={a+broot8: a,b are integars}
a) Prove that S forms a ring.

b) If two members of this ring multiply to make an integer which is a prime number, what form will the factors take?

c) Find two numbers which are prime in the integers but which are composite in this ring and factorise them in this ring.

d) Find two distinct members of this ring which multiply together to make 1.

e) Find as many units as you can in this ring. Explain how you know that they are units.

[aznkid66]

Yeah, the reason I haven't replied yet is that I haven't learned the material, and was hoping a better man could reply so I don't accidentally lead you down the wrong path. So that leads me to...

Disclaimer: I have no clue what I'm talking about, and thus will be making up stuff, be it concepts or terms for concepts. I'll also be leaving out quite a bit when proving. Make sure you ask teachers, classmates, etc. for an explanation so you acquire the language and process.

(a)

From what I can gather, a "ring" is a set with two binary functions, "+" and "•", where (S,+) is an abelian group under addition and (S,•) is a monoid group under multiplication. Since there's no reason not to, I would define (S,+) and (S,•) as "normal" addition and multiplication (for example that of integers).

Associativity/commutativity of (S,+) and (S,•) follows because all elements of S are real numbers and addition/multiplication of real numbers follow associativity/commutativity. Same with the distributive laws.

The four criteria that have yet to be proven are:
- Closure under addition
- Existence of additive identity
- Existence of additive inverse
- Closure under multiplication

Closure under addition:
x,y is an element of S
x = m+nsqrt(8), m and n are integers
y = p+qsqrt(8), p and q are integers
Can x+y always be put in the form a+rootb where a and b are integers?
x+y = (m+p)+(n+q)sqrt(8)
m+p is an integer because m and p are integers and there is closure under addition for the set of all integers.
same for n+q
Therefore x+y can always be put in the form a+bsqrt(8)
Thus, x+y is always an element in S, and there is closure under addition.

Existence of additive identity:
In "normal" addition, the identity is 0. So is 0 and element in S? Yes, when a=b=0.
Thus, there exists an additive identity.

Existence of additive inverse:
Given any element x in S, does there exist a -x such that x + -x = -x + x = 0 and -x is an element in S?
x is an element of S
x = m+nsqrt(8), m and n are integers
Assume -x = -m-nsqrt(8)
Is -x an element of S? Yes, -m and -n are integers if m and n are integers?
Is x + -x = -x + x = 0? Yes.
Thus, there exists an additive inverse.

Closure under multiplication:
x,y is an element of S
x = m+nsqrt(8), m and n are integers
y = p+qsqrt(8), p and q are integers
x*y = (m+nsqrt(8))(p+qsqrt(8)) = (mp+8qn)+(mq+np)sqrt(8)
Are mp+8qn and mq+np integers? Yes, because m, p, 8, q, and n are all integers and there is closure under multiplication and addition for the set of all integers.
Therefore, x*y can always be put in the form a+bsqrt(8)
Thus, x*y is always an element of S, and there is closure under multiplication.

[Maths boy]

Thanks man, I have finally figured out how to do it:-), still some things i need help with.

Below is the group table for the symmetries of a regular hexagon, (clockwise rotations of 60o, 120o, 180o, 240o and 300o, reflections in the mid-points of opposite sides and reflections in the lines through opposite vertices but not in this order)
A B C D F G H J K L M N

A C G J M B K A D N F H L

B F H L K A M B N D C G J

C J K D H G N C M L B A F

D M L H C N F D A B K J G

F L M N G H D F K J A B C

G B A F N C H G L M J K D

H A B C D F G H J K L M N

J D N M A K L J H F G C B

K G C B L J A K F H D N M

L N D K B M J L G C H F A

M H F A J L B M C G N D K

N K J G F D C N B A M L H

a) Find the identity
b) Find the inverse of each element
c) Find the order of each of A, J and N
d) State with reasons which of the elements are reflections and which are rotations. For the ones which are rotations describe which rotation is represented by which element. For the ones which are reflections you are not asked to give any details. Note there is more than one correct answer for this question.
e) State the possible orders of the proper subgroups.
f) Find a subgroup of each of the possible (proper) orders and draw up their group tables.
g) For each subgroup decide if it is cyclic and /or Abelian
h) For the subgroup of order 3, find its left cosets and its right cosets. Is the subgroup normal?

[sputum]

Let's start with a

what does the identity element do?

[Maths boy]

(Original post by sputum)
Let's start with a

what does the identity element do?

The Identity element is an operator that leaves unchanged the element on which it operates.

So if e*x=x
e is the identity

[sputum]

(Original post by Maths boy)
The Identity element is an operator that leaves unchanged the element on which it operates.

So if e*x=x
e is the identity

so we look for an element that has this property (ie the row is the same as the top row and the column is the same as the first column)

Hopefully you can use the inverse property to complete b but if not just say.

Group theory is quite nice because every group-theoretic thing you might need to know is in the Cayley table or can be derived from it, and lots of things like this are available just from inspection but it can be frustrating getting used to it.

Post your answers to any bits you can do if you want them checked, if there's anything in a question part you haven't come across before we can translate it. I love group theory

[Maths boy]

I cant see the inverse property to complete b, sorry:/

[aznkid66]

(Original post by Maths boy)
I cant see the inverse property to complete b, sorry:/

If there exists element x such that x' is the inverse of x, then x*x'=e (where e is the identity). Since you know which element is the identity, find that element in the product-space, and the corresponding further-factor (top) should be the inverse of the corresponding nearer-factor (left). This method should allow you to find the inverses of all 12 elements.

[sputum]

(Original post by Maths boy)
I cant see the inverse property to complete b, sorry:/

No worries. It can take a lot of getting used to.

Having done part a you can now find the identity of any group just by eyeballing its table. Aznkid's method should enable you to find the inverse of any element in any group table once you've identified the identity (just say if you need more on part b btw)

[Maths boy]

Sorry for the late reply, i was figurering out the answer, i think ive got the hang of it. I have a Modelling question that i need help with.

The Pit and the Pendulum
The main aims of this question are to assess the following skills and techniques:
• use of power regression;
• algebraic manipulation;
• communication of ideas;
• commenting on experimental difficulties;
• describing the real-world situation from the shape of a graph.
• identifying your modelling states

A simple pendulum consists of a lead bead hanging from a fixed point at the end
of a string of length L (metres). When the bead is moved slightly to one side and
then released, it will swing to and fro. The time taken for the bead to perform a
complete oscillation (i.e. to return to its starting point) is known as the period, T
(seconds).



A group of students perform an experiment, timing small oscillations of such a
simple pendulum, and obtain the following data.

Length L (metres) 0.6 0.7 0.8 0.9 1.0
Period T (seconds) 1.55 1.68 1.79 1.90 2.01

(a) This part of the question asks you to find a relationship of the form T = aLb between L and T.
Enter the data into your spreadsheet, and use a power regression facilities to find the regression equation.

(i) Write down the values of a, b and r (the correlation coefficient) given by the calculator or spreadsheet.
(ii) Write down the equation for T in terms of L, expressing the constants to
2 decimal places.
(iii) Imagine that you are in the group of students collecting the data. Explain,
as if to a fellow student, two difficulties which you would anticipate
in collecting accurate data.

(b) Theory predicts that, for small oscillations, the formula

gives a good approximation to the relationship, where g is a constant (the
acceleration due to gravity).

This part of the question asks you to compare your regression equation with the formula, and decide if the value of g predicted by the students' results is reasonable.
(i) Rewrite the formula in the form T = aLb, where a and b are
constants. What values of a and b does the formula suggest?

(ii) Compare your values of a and b from part (a)(i) with the expression in
part (b) (i), what do you conclude?

(iii) Rearrange this formula to make g the subject. Hence find the value of
g to 3 significant figures. Compare this value and comment on how reasonable it is, in the light of your answers to part (a)

All throughout your solution identify which of the modelling states you are in.

[Maths boy]

Also can i get help with, this question:

P=
(1234567
3125746)

q=
(1234567
5641723)

r=
(1234567
7631425)

s=
(1234567
5713264)

Find pq(^-1)sr(^2) write the answer as a product of disjoint cylcles. Find its order Parity and Inverse.

[Maths boy]

Not sure if anyone would be able to help with this question:

ALL the following questions require the computer programmes Cabri or Geogebra.

Produce a macro that will create the fractal tree in figure 1 from a line segment



Figure 1

Consider the following picture of the Sydney Harbour Bridge (figure 2)


What is the equation of the top of the bridge? What are the dimensions of the bridge (height from the water, etc)? How accurate are the dimensions you have worked out? Can you find a picture to use that would give you more accurate dimensions?

Create a garage door that can open and close.

Investigate the following geometry theorems using either Cabri or Geogebra:
(a) Pythagoras’ theorem;
(b) The midpoint quadrilateral theorem


Consider the following problem which is a variation of the spider and fly Canterbury puzzle


Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 2 feet from the ceiling, as at A; and a fly is on the opposite wall, 2 feet from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly

[sputum]

(Original post by Maths boy)
Also can i get help with, this question:

Find pq(^-1)sr(^2) write the answer as a product of disjoint cylcles. Find its order Parity and Inverse.

I prefer them like this (let me know if you can't convert between the two)

s=(152743)
P=(132)(4576)
q=(15734)(26)
r=(1754)(26)

Can you confirm if you mean (pq)^-1(sr)²? Can't be too careful

[Maths boy]

(Original post by sputum)
I prefer them like this (let me know if you can't convert between the two)

s=(152743)
P=(132)(4576)
q=(15734)(26)
r=(1754)(26)

Can you confirm if you mean (pq)^-1(sr)²? Can't be too careful

yes thats exactly what i meant

[Maths boy]

(Original post by aznkid66)
If there exists element x such that x' is the inverse of x, then x*x'=e (where e is the identity). Since you know which element is the identity, find that element in the product-space, and the corresponding further-factor (top) should be the inverse of the corresponding nearer-factor (left). This method should allow you to find the inverses of all 12 elements.

I think i get it now, I get the inverse as H as it does not change anything it multiplies. just to confirm i am finding the inverses right, is the inverse of a, m?

[Maths boy]

(Original post by sputum)
No worries. It can take a lot of getting used to.

Having done part a you can now find the identity of any group just by eyeballing its table. Aznkid's method should enable you to find the inverse of any element in any group table once you've identified the identity (just say if you need more on part b btw)

Done part a, and pretty sure ive done part b, what about part c and d?