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Topic: A factorization problem (Read 440 times) |
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Benny
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A factorization problem
« on: Jun 15th, 2009, 2:44pm » |
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True or false?
(1) x4 - x2 is a multiple of 12 for every integer value of x.
(2) x5 - x3 is a multiple of 24 for every integer value of x ?
For (1):
x4 - x2 = xx(x2 - 1) = xx(x - 1)(x + 1) = x(x - 1)x(x + 1) ..... (1)
we know that the product of three consecutive numbers is always divisible by 3.
That is, (x-1)x(x+1) is always divisible by 3.
If Eq.#1 is divisible by 12, that is
If x(x - 1)x(x + 1) is divisible by 4 and 3
If I try a few sets of integers, I find it to be true.
But, I'm looking for a formal proof
For (2)
x5 - x3 = xxx(x2 - 1) = xx(x - 1)x(x + 1)
I got stuck here
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towr
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Re: A factorization problem
« Reply #1 on: Jun 15th, 2009, 3:16pm » |
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on Jun 15th, 2009, 2:44pm, BenVitale wrote:
But, I'm looking for a formal proof |
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? You have a factor of 3, and two even factors: either x is even, which occurs twice, or x is odd, in which case x-1 and x+1 or even.
3*2*2=12, what more formality do you need?
Quote:either x is even, in which case it accounts for 3 even factors; or if x is odd, then either x-1 or x+1 is a multiple of 4, while the other is a multiple of 2.
So together there must be a factor of 24.
And in either case, you can suffice by finding the residues modulo 12 and 24 respectively for the numbers 1..12 and 1..24 respectively, if they're all 0 then they are 0 for all x.
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| « Last Edit: Jun 15th, 2009, 3:20pm by towr » |
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Benny
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Re: A factorization problem
« Reply #2 on: Jun 15th, 2009, 5:46pm » |
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I wanted to write a rigorous proof ... I missed a step somewhere. Could you tell me what I did miss here?
In #1
x4 - x2 as xx(x2 - 1)
The eq. is divisible by 3, but is the eq. also divisible by 4?
There are 2 cases:
(a) x is even, that is x = 2k
(b) x is odd, that is x = 2k + 1
If x is even, then
xx = (2k)(2k)
this shows that xx is divisible by 4.
the expression is divisible by 4.
If x is odd, then
(x - 1)(x + 1) = 2k(2k + 2) = 4k(k + 1)
this shows that the product is a multiple of 4.
The expression is x(x - 1)x(x + 1)
If x even:
4kk(2k - 1)(2k + 1)
this shows that it is also divisible by 3.
Hence the expression is divisible by 12.
If x odd:
(2k + 1)(2k + 1)4k(k + 1)
this expression is divisible by 4. But I don't see how it is also divisible by 3.
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| « Last Edit: Jun 15th, 2009, 5:48pm by Benny » |
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towr
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Re: A factorization problem
« Reply #3 on: Jun 16th, 2009, 12:35am » |
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on Jun 15th, 2009, 5:46pm, BenVitale wrote:If x odd:
(2k + 1)(2k + 1)4k(k + 1)
this expression is divisible by 4. But I don't see how it is also divisible by 3. |
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You had already established it was divisible by 3; but if you want to do it again,
(2k + 1)(2k + 1)4k(k + 1) = (2k + 1) * 2k*(2k + 1)*(2k + 2)
2k, (2k + 1) and (2k + 2) are three consecutive integers, hence one is divisible by 3.
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Benny
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Re: A factorization problem
« Reply #4 on: Jun 16th, 2009, 9:26am » |
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It was there in front of me... and I didn't see it.
Thanks, towr.
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