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MGMAT smallest prime factor - Need expert help

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by voodoo_child » Mon May 07, 2012 8:37 am
X = (3 * 5 * 7 ....47) {all +ve odd integers}

Y= X + 2

what's the smallest prime factor of Y?

I know that it will be greater than 47. I am not sure how to calculate 53.

Any help?

OA = 53
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by voodoo_child » Mon May 07, 2012 1:40 pm
Can any of the experts please help me? Thanks
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by Bill@VeritasPrep » Mon May 07, 2012 1:58 pm
Where is the question from, and what were the answer choices?
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by voodoo_child » Mon May 07, 2012 2:18 pm
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
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by Bill@VeritasPrep » Mon May 07, 2012 2:23 pm
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.
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by voodoo_child » Mon May 07, 2012 2:35 pm
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?
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by Bill@VeritasPrep » Mon May 07, 2012 2:42 pm
Well, the smallest prime number greater than 50 is 53.
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by voodoo_child » Mon May 07, 2012 2:59 pm
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?
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by Bill@VeritasPrep » Mon May 07, 2012 3:05 pm
Doesn't the explanation say that 53 is the minimum possible prime factor?
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by dabral » Mon May 07, 2012 9:57 pm
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
Free Video Explanations: 2021 GMAT OFFICIAL GUIDE.
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by mathbyvemuri » Tue May 08, 2012 1:26 am
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.
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