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chuckisnutz
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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
The answer is (E)
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?
#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
The answer is (E)
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?
