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Factor of a sum

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Manager
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Factor of a sum [#permalink]

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New post 13 Jun 2009, 12:24
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Hi there,

I always have a hard time to find a particular factor of a sum:

For instance, X = 20! + 16

I know that every number from 1 to 20 is a factor of 20!. But as soon as another number is added, this nice statement does not hold anymore, right?

So can someone explain (1) why 20 is no longer a factor of this sum and (2) why 16 is still a factor?

Thanks!

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Re: Factor of a sum [#permalink]

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New post 13 Jun 2009, 18:36
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I guess because adding is not multiplying. (Multiplying is in sense adding, but that's a different story).So if you would multiply by 16 than all the factors would stay the same. :)
Let me try to reason it out with you. But don't take my word for granted. And excuse me if I do it primitively.

Ok. First of all, when we add number we get a different number with different factors.For example factorise 100 -- 1, 2, 4, 5, 10, 20, 25, 50, 100, now 100+1 - is prime - it has only two factors 1 and 101

To answer your question. So the number X consist of factors \(f_1* f_2 *f_3*f_4... f_n\) when we add another number there are two (I guess) possibilities (I) the number we add may may have common factors with number X, or (II) may not have common factors with number X, (besides 1 of course ) ( I hope I won't go too far in reasoning)

So in first case I guess essentially what is happening is number X with factors \(f_1* f_2 *f_3*f_4... f_n\) is addied with another number with factors lets say \(f_2*f_3\) which is divisible by \(f_2\) (or \(f_3\))because \(\frac{ f_1* f_2 *f_3... f_n+f_2*f_3}{f_2}\) -> \(\frac{f_2(f_1*f_3... f_n+f_3)}{f_2} = f_1*f_3... f_n+f_3\)
(This answers you question what factors - 20! +16 has. Common factors - 1, 2, 4, 8, 16 . So 20 is not a factor while 16 is :-D ))

Hope I didn't mess up. Does that answer your questions?
Should we talk about a second case when there are no common factors?
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New post 13 Jun 2009, 18:58
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Thanks man. I figured it out in the meantime:

If 2 numbers have common factors, these factors carry over when the two numbers are added (or subtracted or multiplied):

So 20! and 16 have the following common factors: 1,16,2,8,4. Therefore, the sum of 20! and 16 has the same factors (1,16,2,8,4).

Let's take another example: 3 and 5 only have one common factor: 1. The sum of 3 and 5 is 8, which is divisible by 1, 2, 4, and 8.

So I would assume that the sum of 20! and 16 has the factors 1,16,2,8,4 PLUS a certain number of other factors, right?

I will read your explanation tomorrow. Too tired to process all this information at 4am!

Thanks again
Steve

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Re: Factor of a sum   [#permalink] 13 Jun 2009, 18:58
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Factor of a sum

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