MGMAT smallest prime factor - Need expert help

Can any of the experts please help me? Thanks

Where is the question from, and what were the answer choices?

Sure, Bill - The question is from Manhattan GMAT.
Answer choices were -

1) Y>50
2) 30 <= y <= 50
3) 10 <= Y <= 30
4) 3<= y <10
5) y=2

MGMAT says that y = 53 is the minimum possible prime factor of (3...47)+2

The trick here is to realize that any of the prime numbers that fall between 3 and 47 (inclusive) are already factors of 3#47. As a result, they cannot also be factors of 3#47 + 2. 48, 49, and 50 are not primes, so y must be greater than 50.

The trap answer here is 2, I believe. It's conspicuously absent from 3#47, so it could be tempting. Since # represents the multiplication of odd integers only, we know that 3#47 = 3*5*7*...*49, which will be an odd product. An odd plus an even gives us an odd sum, so (3#47) + 2 is odd. Therefore, 2 cannot be a factor.

I've seen variations on this question; one said that x was the product of all even integers from 2 to 50, then asked for the smallest prime factor of x + 1.

Bill@VeritasPrep wrote:The trick here is to realize that any of the prime numbers that fall between 3 and 47 (inclusive) are already factors of 3#47. As a result, they cannot also be factors of 3#47 + 2. 48, 49, and 50 are not primes, so y must be greater than 50.

The trap answer here is 2, I believe. It's conspicuously absent from 3#47, so it could be tempting. Since # represents the multiplication of odd integers only, we know that 3#47 = 3*5*7*...*49, which will be an odd product. An odd plus an even gives us an odd sum, so (3#47) + 2 is odd. Therefore, 2 cannot be a factor.

I've seen variations on this question; one said that x was the product of all even integers from 2 to 50, then asked for the smallest prime factor of x + 1.

Sure, Bill. I got this one correct. But why does OE say that minimum prime factor = 53?

Well, the smallest prime number greater than 50 is 53.

Bill@VeritasPrep wrote:Well, the smallest prime number greater than 50 is 53.

ok. But how do I know whether it's a factor of y? Can you please help me?

Doesn't the explanation say that 53 is the minimum possible prime factor?

What Bill is saying is that we can only conclude that the smallest prime factor of y must be greater than 47, for all we know it could be 53(least prime number greater than 47), or it could be 89, or Y itself could be prime. We have no way of knowing that based on the expression given for Y.

This question is based on a difficult GMATPrep question that Bill alluded to, Here is the link:
https://www.beatthegmat.com/functions-t85150.html#377237

Another problem similar in concept but a little bit easier is Problem Solving#77 from Official Guide GMAT 13th Edition.

Cheers,

Dabral

It's a good problem. A simple logic breaks the puzzle out.
consider a small example here:
3*11 + 2 = 35 is divisible by 7,but 3*7 + 2 = 23 is not. What can be inferred from this?
First case: (3*11 + 2)
The two parts 3*11 and 2 are not divisible by 7 individually, but the sum does.
Second case: (3*7 + 2)
As 3*7 is already a factor of 7, the added value must also be divisible by 7, for the final sum is to be divisible by 7. But the added value 2 is not divisible by 7 and hence the whole part does not.
The basic funda here is - "if X is divisible by p and Y is divisible by p then, X+Y is also divisble by p".

Now, come back to our problem.
(3*5 + 2) is not divisible by 3 and 5 as the second part '2' is not divisible by them.
similarly (3*5*7 + 2) is not divisible by any of 3,5, and 7.
For (3*5*7 + x) is to be divisible by 3, x must be 3 or multiple of 3. Similar logic is applicable for the divisibility by 5 and 7.

As '2' is a number which is not divisible by any of the odd numbers 3,5,7,...47,
the number (3*5*7*...47)+2 is not divisible by any of 3,5,7,...47.
And so the minimum possible prime number to be considered for divisiblity is 53.