yellowho wrote:Is the prime number q equal to 29 ?
(1) (q - 1) has 6 factors
(2) (q + 1) has 2 and 3 as prime factors

Note that, 29 satisfy both the statements.
Hence, if we can find at least one example that satisfy any of the statement, then the statement will be insufficient.

Statement 1: There can be infinite number of integers with 6 factors. Hence, infinite possible values of q are there.

Not sufficient

Statement 2: There can be infinite number of integers with 2 and 3 as prime factors, i.e. divisible by 6. Hence, infinite possible values of q are there.

Not sufficient

1 & 2 Together: Both statements together also doesn't limit the number of possible values of q. This is because both of them is also not an unique property for a particular integer. For example, q can be equal to 29 as well as 53.

Not sufficient

The correct answer is E.

pesfunk wrote:HI Anurag, is there a quick way to get 53?

There is some logic I can give you which will show you the way, but it is not definitely quick. In my opinion, this particular problem is not asking for a solution. It is testing whether you have the sense to identify a statement as unique property for an integer or not.

Also knowing the fact that there is no particular formula or relation by which we can uniquely point out prime numbers, is sufficient to answer the problem.

Now, here is the logic I was talking about.
Disclaimer: I am going to make some huge step jumps so if you do not follow do not bother. This is not really necessary.

Note that for statement 1 to be true, the prime factorization of (q - 1) must be of the form a²b, where a and b are prime numbers. And for statement 2 to be true, (q + 1), i.e (a²b + 2) must be divisible by 6, i.e. both by 2 and 3. Now it can be shown that a will be equal to 2 and b will be greater than 3. Therefore, q will be of the form (a²b + 1) = (4b + 1), where b is any prime number greater than 3.

Again it can be shown that, q will of the form (24n + 5) for some positive integer n. (Note that this doesn't mean any integer of the form (24n + 5) will be a possible value of q.)

Now you have to look for prime integers of the form (24n + 5).
For, n = 1 => q = 29
For, n = 2 => q = 53
...

Again, this doesn't mean all integer of the form (24n + 5) will be a possible value of q. For example, n = 3 doesn't give a prime.

clock60 wrote:....

do you think this approach is valid or it is just oversimplification
thanks

Yes, you're reasoning is good.
We can generalize it more and get a general expression for such primes as I did in one of the above posts.

But, none of this is required (even picking any number), if you know that there is no unique way to represent prime numbers unless they point to a particular prime. Therefore, any bunch of properties can point to more than one prime unless they limit the range.