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# Prime Factorization of Consecutive Integers

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Intern
Joined: 08 Apr 2015
Posts: 10
Prime Factorization of Consecutive Integers  [#permalink]

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01 Jul 2015, 00:50
1
Hello everybody,

lets assume we have 6 consecutive integers multiplied, for instance (n-3)(n-2)(n-1)n(n+1)(n+2). Now I want to perform a prime factorization in order to determine possible factors of this expression. Of course I cannot determine all factors but at least some of which I know for sure that they must be included.

Since we have 6 consecutive integers, at least one number within that set must be divisible by 1,2,3,4,5, and 6. Next, I wanted to prime factorize these numbers:
1 = 1
2 = 2
3 = 3
4 = 2*2
5 = 5
6 = 2*3

As you can see, some of the prime factors overlap. In total we have:
1,2,2,2,2,3,3,5

Can I conclude that all the above stated factors are included in the set or do I have to eliminate the overlaps? If all the factors are included, then any combination of these factors must also be a factor of the expression, for instance: 2*2*2*3 = 24 must be a factor of (n-3)(n-2)(n-1)n(n+1)(n+2)
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Posts: 1811
Re: Prime Factorization of Consecutive Integers  [#permalink]

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01 Jul 2015, 02:09
1
nicok06 wrote:

Can I conclude that all the above stated factors are included in the set

Yes, you can - you don't want to eliminate the overlaps. The general rule is this: the product of k consecutive positive integers is always divisible by k!

So, for example, if you multiply six consecutive integers, the product will always be divisible by 6! = 6*5*4*3*2*1, and of course by any factor of 6! as well.

The reason this is all true is because multiples are equally spaced - since multiples of 6 are 6 apart, then in any list of six consecutive integers, we must always have exactly one multiple of 6. Similarly, since multiples of 5 are 5 apart, we always must have at least one multiple of 5 (and possibly two of them) among any six consecutive integers, and so on.
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Re: Prime Factorization of Consecutive Integers  [#permalink]

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29 Jan 2018, 00:19
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Re: Prime Factorization of Consecutive Integers   [#permalink] 29 Jan 2018, 00:19
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