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Number properties | Prime factors | Advanced, 700-800

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Number properties | Prime factors | Advanced, 700-800

by ChessWriter » Thu Mar 29, 2012 9:00 pm
If a and b are odd integers, a#b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3#47)+2, which of the following must be true?

A) y>50
B) 30 <= y <=50
C) 10 <= y <= 30
D) 3 <= y <= 10
E) y=2

(<= stands for the 'greater than or equal to' function)

The Official answer to this question from the Manhattan Advanced Quant strategy supplement is A. However, I could not understand the explanation they've provided. It looks completely counter-intuitive to me that y>50 could be the right answer.

Can anyone please help?
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by Bill@VeritasPrep » Thu Mar 29, 2012 9:16 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.

Bill
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by GMATGuruNY » Fri Mar 30, 2012 3:08 pm
ChessWriter wrote:If a and b are odd integers, a#b represents the product of all odd integers between a and b, inclusive. If y is the smallest prime factor of (3#47)+2, which of the following must be true?

A) y>50
B) 30 <= y <=50
C) 10 <= y <= 30
D) 3 <= y <= 10
E) y=2

(<= stands for the 'greater than or equal to' function)

The Official answer to this question from the Manhattan Advanced Quant strategy supplement is A. However, I could not understand the explanation they've provided. It looks completely counter-intuitive to me that y>50 could be the right answer.

Can anyone please help?
I received a PM asking me to comment.

Since 3#47 is the product of all of the odd integers between 3 and 47, inclusive, 3#47 is odd.
Thus, 3#47 and (3#47)+2 are consecutive odd integers.
Consecutive odd integers are COPRIMES: integers that share no factors other than 1.
Thus, 3#47 and (3#47)+2 are COPRIMES: they share no factors other than 1.

Every prime number between 3 and 47, inclusive, is a factor of 3#47.
Thus NONE of the prime numbers between 3 and 47, inclusive, can be a factor of (3#47)+2 (since consecutive odd integers are coprimes).

Since an odd integer such as (3#47)+2 cannot have a factor of 2, and since none of the prime numbers between 3 and 47, inclusive, is a factor of (3#47)+2, the value of y -- the smallest prime factor of (3#47)+2 -- must be greater than 50.

The correct answer is A.

For a similar problem about coprimes, check here:

https://www.beatthegmat.com/functions-t83704.html
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by bubbliiiiiiii » Mon Apr 02, 2012 9:41 pm
How did you conclude this?
Since 3#47 is the product of all of the odd integers between 3 and 47, inclusive, 3#47 is odd.
Regards,

Pranay
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by GMATGuruNY » Mon Apr 02, 2012 9:43 pm
bubbliiiiiiii wrote:How did you conclude this?
Since 3#47 is the product of all of the odd integers between 3 and 47, inclusive, 3#47 is odd.
odd*odd = odd.
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