x is the product of all even numbers from 2 to 50, inclusive

Hi All,

We're told that X is the product of all even numbers from 2 to 50, inclusive. We're asked to define what the smallest prime factor of (X+1) must be. This particular question is a variation on an Official question that periodically pops up in the forums. The main idea behind this prompt is:

"The ONLY number that will divide into X and (X+1) is 1."

In other words, NONE of the factors of X will be factors of X+1, EXCEPT for the number 1.

Here are some examples:
X = 2
X+1 = 3
Factors of 2: 1 and 2
Factors of 3: 1 and 3
ONLY the number 1 is a factor of both.

X = 9
X+1 = 10
Factors of 9: 1, 3 and 9
Factors of 10: 1, 2, 5 and 10
ONLY the number 1 is a factor of both.
Etc.

Since X is (50)(48)(46)....(4)(2)....we can deduce....
1) This product will have LOTS of different factors
2) NONE of those factors will divide into (X + 1) except for the number 1.

X contains all of the primes from 2 through 23, inclusive (the 23 can be "found" in the "46"), so NONE of those primes factors will be in (X + 1). We don't have to calculate the value of the smallest prime factor though ; we know that it MUST be a prime greater than 23....and there's only one answer that fits.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

AAPL wrote:Veritas Prep

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

x = (2)(4)(6)....(46)(48)(50)
= (1)(2)(2)(2)(3)(2).....(23)(2)(24)(2)(25)(2)

Notice that:
x is divisible by 2. This tells us that x+1 is 1 greater than a multiple of 2. In other words, x+1 is NOT divisible by 2
x is divisible by 3. This tells us that x+1 is 1 greater than a multiple of 3. In other words, x+1 is NOT divisible by 3
x is divisible by 4. This tells us that x+1 is 1 greater than a multiple of 4. In other words, x+1 is NOT divisible by 4
x is divisible by 5. This tells us that x+1 is 1 greater than a multiple of 5. In other words, x+1 is NOT divisible by 5
.
.
.
x is divisible by 23. This tells us that x+1 is 1 greater than a multiple of 23. In other words, x+1 is NOT divisible by 23
x is divisible by 24. This tells us that x+1 is 1 greater than a multiple of 24. In other words, x+1 is NOT divisible by 24
x is divisible by 25. This tells us that x+1 is 1 greater than a multiple of 25. In other words, x+1 is NOT divisible by 25

We see that x+1 is NOT divisible by 2 to 25
In other words, all integers from 2 to 25 are NOT factors of x+1

So, if a number IS a factor of x+1, that number must be greater than 25

Answer: E

Cheers,
Brent

AAPL wrote:Veritas Prep

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

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.

Answer: E