## Nasty Little Veritas Factorial Question

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### Nasty Little Veritas Factorial Question

by chuckisnutz » Thu Nov 17, 2011 5:54 pm
Hi there! My understanding of factorials is weak, so when I rolled across this question in my Veritas Prep Arithmetic book, I was stumped.

#82
x is the product of all even numbers from 2 to 50, inclusive. The smallest prime factor of x + 1 must be:

A) Between 1 and 10
B) Between 11 and 15
C) Between 15 and 20
D) Between 20 and 25
E) Greater than 25

Here is the explanation given in the book:
The product of all even numbers between 2 and 50 can be written as (2^25)*(1*2*3*...*25), because a 2 can be factored out from all of the 25 even numbers that lie between 2 even numbers that lie between 2 and 50, inclusive. In other words, all the numbers from 1 to 25 inclusive are factors of x. The important number property to realize here is that if a number other than 1 is a factor of x + 1. In this Problem you can be sure that all numbers between 1 and 25 inclusive are factors of x and therefore none of them (except for 1) can be factor of x + 1. Therefore the first prime number that could possibly divide into x + 1 has to be greater than 25. This is a very difficult question but very similar problems can be found on the GMAT.
/end of plagiarism

So, my question is, why is it that when you add 1 to 50!, that suddenly 1 through 25 can't be factors anymore?

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by vineeshp » Thu Nov 17, 2011 6:34 pm
Well cos adding does not mean factors carry over.
For eg. 1 + 25 (Factors 1 and 5) end up in 26 factors (2 and 13).

1 and 2 add up to 3 but 3 does not have any factors of 1 and 2

4 and 6 add up to 10 but 10 factors into 2 * 5. So there is no such concept.
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by pemdas » Thu Nov 17, 2011 7:39 pm
absolutely agree with Vineesh, to add up more the factorial order 1...25 doesn't carry out the same factors amongst 1~25, the only one is 1 and this isn't prime.

But be sure the new number must be divisible by a prime as any number (!) is divisible by prime except for 1 when we perform prime factorization. So either we have a prime > 25 or the prime number divisor of x+1 is the number itself, i.e. x+1 (and x+1 will be prime then)
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by GMATGuruNY » Thu Nov 17, 2011 8:00 pm
Consecutive integers are COPRIMES: they share no factors other than 1. I posted an explanation here:

https://www.beatthegmat.com/tough-prime- ... 25475.html
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by Scott@TargetTestPrep » Thu Oct 25, 2018 8:16 am
chuckisnutz wrote:Hi there! My understanding of factorials is weak, so when I rolled across this question in my Veritas Prep Arithmetic book, I was stumped.

#82
x is the product of all even numbers from 2 to 50, inclusive. The smallest prime factor of x + 1 must be:

A) Between 1 and 10
B) Between 11 and 15
C) Between 15 and 20
D) Between 20 and 25
E) Greater than 25
Two consecutive integers do not share any common prime factors. Thus, we know that x and x + 1 cannot share any of the same prime factors.

We also see that x, the product of the even numbers from 2 to 50, contains prime factors of 2, 3, 5, 7, 11, 13, 17,19, and 23.

Thus, since x contains the primes from 2 to 23, we see that the smallest prime factor of x + 1 must be at least 29, i.e., greater than 25.