Veritas Aritmetic Q

[Goldfinger2001]

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

OA: e

I got the OA but I don't get the explaination.

It says the product of all even numbers between 2 and 50 can be written as 2^25*(1*2*3...*25)
In my opinion it should say the product...can be written as 2*(1*2*3*4*5*6...25)...and because of that the smallest possible PF must be greater than 25.

Am I thinking wrong? I just get confused with the 25th power, thats all.

Thx

[Geva@EconomistGMAT]

Your version would be correct if we were disucssing sums, not products:
2(1+2+3...+25) is indeed equal to 2*1+2*2+2*3+2*4...+2*25

In the case of a product, the parentheses don't actually mean anything:
2*(1*2*3*4*5...) is just the same as 2*1*2*3*4*5... and not what you're looking for, which is 2*4*6*8..*50.

As to why 2^25, is indeed correct: break down each of the prime numbers in the product into 2*something: 2 is (2*1), 4 is (2*2), 6 is (2*3), etc.
Thus, instead of 2*4*6*8*...48*50, write
(2*1)*(2*2)*(2*3)*(2*4)*...(2*24)*(2*25)
Get all the 2s out, and you are left with 2*2*2*2*...(1*2*3*4...*24*25), or 2^25*(1*2*3*4...*24*25).

[Geva@EconomistGMAT]

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

OA: e

I got the OA but I don't get the explaination.

It says the product of all even numbers between 2 and 50 can be written as 2^25*(1*2*3...*25)
In my opinion it should say the product...can be written as 2*(1*2*3*4*5*6...25)...and because of that the smallest possible PF must be greater than 25.


Am I thinking wrong? I just get confused with the 25th power, thats all.

Thx

And btw, the powers of 2 aren't important for the question. What Veritas is trying to show in this step (and what you need to realize for the GMATPREP question this Q is based on), is that the product of all the even integers will be divisible by all the prime numbers up to 25, just because it will include the even multiples of all of these primes. So if x is divisible by all primes up to 25, (in other words, if x is a multiple of all primes up to 25), then x+1 will not be any of the primes up to 25 - thus indicating that whatever the smallest prime factor of x, it must be greater than 25.

[Goldfinger2001]

Thanks Geva,

I somehow didn't realize that 2*(1*2*3..) is the same as 2*1*2*3...
too early I guess

I understood that x+1 must have a prime bigger than 25 though, I just got stuck with the calculation above.