Integer x is equal to the product of all even numbers...

[AAPL]

Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x-1, then which of the following expressions must be true?

$$A.\ \ 0\ <\ y\ <\ 4$$
$$B.\ \ 4\ <\ y\ <\ 10$$
$$C.\ 10\ <\ y\ <\ 20$$
$$D.\ 20\ <\ y\ <\ 30$$
$$E.\ \ y\ >\ 30$$

The OA is E.

Is there a strategic approach to this question? Can any experts help? Thanks.

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Hi AAPL,

We're told that X is equal to the product of all EVEN numbers from 2 to 60, inclusive and that Y is the smallest PRIME number that is also a factor of (X-1). We're asked which of the following expressions MUST be true.

This question is a great 'concept' question, meaning that you don't have to do much math to answer it IF you recognize the concept(s) involved. The concept behind this question is - "when X is an integer, the ONLY factor that X and (X+1) have in common is 1." In other words, NONE of the factors of X are factors of (X+1) EXCEPT for the number 1.

For example:
Factors of 2: 1 and 2
Factors of 3: 1 and 3
The only common factor is 1

Factors of 27: 1, 3, 9 and 27
Factors of 28: 1, 2, 4, 7, 14 and 28
The only common factor is 1

From the prompt, we know that X will be a huge even number (since it's the product of all of the even numbers from 2 to 60, inclusive) and it's also a multiple of a lot of different odd and even numbers (for example, 58 is a factor of X and it can be prime-factored down into 2 and 29) so we know that X is a multiple of 29. By extension, we know that X has all of the PRIME numbers from 2 to 29 as factors, so (X-1) will NOT have any of those numbers as factors. Thus, the smallest prime factor of (X-1) will have to be something greater than 29...

Final Answer: E

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

[Scott@TargetTestPrep]

Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x-1, then which of the following expressions must be true?

$$A.\ \ 0\ <\ y\ <\ 4$$
$$B.\ \ 4\ <\ y\ <\ 10$$
$$C.\ 10\ <\ y\ <\ 20$$
$$D.\ 20\ <\ y\ <\ 30$$
$$E.\ \ y\ >\ 30$$

The OA is E.

Is there a strategic approach to this question? Can any experts help? Thanks.

Since x is the product of all the even numbers from 2 to 60, we see that it contains prime factors from 2 to 29. Since consecutive numbers cannot share any of the same prime factors, x - 1 cannot contain any primes from 2 to 29, so the smallest prime factor of x - 1 must be greater than 30.

Answer: E