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

Hi All,

The above prompt is essentially just a 'lift' of the following GMAC question (but the concept is exactly the same):

For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of H(100) + 1, then p is:

1. between 2 and 10
2. between 10 and 20
3. between 20 and 30
4. between 30 and 40
5. greater than 40

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 the H(100) is (100)(98)(96)....(4)(2)....we can deduce....
1) This product will have LOTS of different factors
2) NONE of those factors will divide into H(100) + 1.

H(100) contains all of the primes from 2 through 47, inclusive (the 47 can be "found" in the "94"), so NONE of those will be in H(100) + 1. We don't even have to calculate which prime factor is smallest in H(100) + 1; we know that it MUST be a prime greater than 47....and there's only one answer that fits.

Final Answer:

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

hi Bunuel
since x is the product of all even integers from 2 to 50 inclusive,

x = (2 * 4 * 6 * 8 * 10 * 12 * 14.....* 50)
which can be rewriten as

2( 1 * 2 * 3 * 4 * 5 * 6 * 7 *........* 25)
so, x is equal to 2 * 25!

please say to me why this is not okay ...

thanks in advance, man

gmatcracker2017:

I believe you are mixing the rule up with a question where we are talking about the sum of all even integers between 2 and 50. Then it would be 2+3+6...+50 which can factor out just one 2, but since it's the product, you have to factor out all 25 "2"s. I believe that is where your thought processes is getting confused, but mine gets confused often, so Bunuel may be best to confirm.

hi zflodeen

oh! I got it ...

I must be sleeping as I posted this ...

thanks

No worries gmatcracker2017. Happens all the time to me!

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

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

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