I picked up some catalogues of math puzzles to do for fun and it's going pretty well so far but now I'm getting to the harder ones. The catalogue claims you don't need anything beyond calculus to do the problems, and that without calculus you should be able to do all but 4 of them. Now I graduated 2 years ago but I really don't remember ever learning how to prove something like X^a + X^b will always produces numbers divisible by c if and only if a and b are certain type of number. I kinda know why it's true, but I don't know how to put it into a proof. So uhh, TL math people, could you explain this or link me to some tutorial on doing this stuff? I just want the basics of how you prove stuff like this.

Vin{MBL}

5185 Posts

April 04 2007 06:50 GMT

hmm you can PM WhatisProtoss... hes a math major if im not mistaken

MathAreHard

United States74 Posts

April 04 2007 06:53 GMT

Goodness gracious...

Hippopotamus

1914 Posts

April 04 2007 06:55 GMT

Yeh... I should have majored in math but my high school had such a shitty program that I actually hated math. I actually got trig/geo/alg for dummies before getting this catalogue and it actually turned out that in several semesters of math that I took in my high school only the algebra was fully covered, half the stuff in the geometry and trig books I was never even taught.

Vin{MBL}

5185 Posts

April 04 2007 06:59 GMT

On April 04 2007 15:53 MathAreHard wrote:
Goodness gracious...

HOLY HELL
ROFLMAO

if there was liquiscoop... post of the YEAR

thedeadhaji

39489 Posts

April 04 2007 06:59 GMT

hmm, I feel like this step:
X^a + X^b -> X^a[ 1 + X^(b-a)]

is the first step..?

thedeadhaji

39489 Posts

April 04 2007 07:04 GMT

X^a + X^b -> X^a[ 1 + X^(b-a)]

if a is greater than b, then X^(b-a) will have a negative exponent, which means that it will be a fraction.

Assuming that "divisible" means "resulting in whole number quotients", and that X is a whole number as well, we cannot allow this to happen.

But on the other hand, if we take out X^b, then we get:
X^b [ X^(a-b) + 1], and if b is greater than a, we are left with a fraction again.

This means that a must equal b for X^a + X^b to be divisible by c.

does this sound right?

edit: wait hmm, even if we get a fraction b/c a is greater than b, if the +1 makes it greater than 1, the multiplication with X^a could make the expression a whole number again...? hmm...

MathAreHard

United States74 Posts

April 04 2007 07:07 GMT

Hey, what is liquiscoop? I only see "liquibet"

zdd

1463 Posts

April 04 2007 07:14 GMT

I think he means specific values for a,b,c (because x^a+x^b will always be divisible by c=x^a+x^b)
sort of like for what values a and b is x^a+x^b divisible by 5?
or even like for what values a,b is 6^a+6^b div. by 5?

Hippopotamus

1914 Posts

April 04 2007 07:16 GMT

#10

Well, yeah... ok that made sense but there're like 50 more of those left and they're not all like that. Thanks, at least I can do a couple of them now. But this is exactly what I struggle with, the reformulation in your first step was fairly basic but it didn't even come across to me to do that first. Another thing is proving divisibility though, for specific numbers.

azndsh

United States4447 Posts

April 04 2007 07:26 GMT

#11

you should give a lot more details... I feel it's a bit too vague to have any idea what you're talking about

Hippopotamus

1914 Posts

April 04 2007 07:39 GMT

#12

Well that's just the general form of a few I encountered, a bunch of others involve proving whether numbers are rational or not. A bunch of them have just symbols. I am given several integers defined by letters and then it's like xy, wz + xy, wz and I need to prove that every listed of multiple of those integers (xy, wz, xy, wz) are divisble by the same number provided that all three expressions (wz + xy counts as one) are divisble by it as well. I'm not sure I even understand haji's proof. It seems to me that X^a times a fraction ie 1+X^(negative number) could sometimes end up as a whole number

Vin{MBL}

5185 Posts

April 04 2007 07:45 GMT

#13

On April 04 2007 16:07 MathAreHard wrote:
Hey, what is liquiscoop? I only see "liquibet"

it was a section where every week there was a news post on the best parts of the forum like notable threads , funny posts, funny pics etc...its been discontinued though

BuGzlToOnl

United States5918 Posts

April 04 2007 08:04 GMT

#14

On April 04 2007 16:45 Vin[MBL] wrote:

Show nested quote +

it was a section where every week there was a news post on the best parts of the forum like notable threads , funny posts, funny pics etc...its been discontinued though

has it really been discountinued or have the admist just been really lazy? :p

Zalfor

United States1035 Posts

April 04 2007 09:07 GMT

#15

i'm majoring in math.

.kaz

1963 Posts

April 04 2007 09:18 GMT

#16

On April 04 2007 18:07 Zalfor wrote:
i'm majoring in math.

Well thats nice and all, but why don't you try to help him?
Yeah, during high school I got up to calculus in math, and after high school I realized that they skipped about half of the courses and some parts were pretty important.

yisun518

Canada480 Posts

April 04 2007 09:26 GMT

#17

i remember part of this from classical algebra, remainder theorem, little fermat theorem.

if u provide a specific question, maybe i can try to dig my past notes up, and see if i can still do them

last time someone asked some remainder problem in a study room, took me 20 min to come up with the proof -_-... it was a lot faster when i was taking the course, guess im getting old