Is the prime number q equal to 29 ?

Hi everyone,

Is the prime number q equal to 29 ?

(1) q - 1 has exactly 6 positive factors.
This is valid for both 29 and 53. Hence insufficient

(2) 2 and 3 are prime factors of q + 1
This is both valid for 29 and 23. Hence insufficient

Taken together the statements are valid for 29 and 53.

Hence E

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have 1 variable (\(q\)) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
We have \(q - 1 = a^5\) or \(q - 1 =a^2 \cdot b^3\) for prime numbers \(a\) and \(b\).
If \(q - 1 = 2^2 \cdot 7^1\), then we have \(q = 29\) and the answer is 'yes'.
If \(q - 1 = 2^2 \cdot 3^1\), then we have \(q = 13\) and the answer is 'no'.
Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
Since \(2\) and \(3\) are prime factors of \(q+1\), \(q+1\) is a multiple of \(6\).
If \(q+1=30\), then we have \(q=29\) and the answer is 'yes'.
If \(q+1=24\), then we have \(q=23\) and the answer is 'no'.
Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
We have \(q - 1 = a^5\) or \(q - 1 =a^2 \cdot b^3\) for prime numbers \(a\) and \(b\).
Since \(2\) and \(3\) are prime factors of \(q+1\), \(q+1\) is a multiple of \(6\).
If \(q - 1 = 2^2 \cdot 7^1\), then \(q+1=30\) is a multiple of \(6\) and we have \(q=29\) and the answer is 'yes'.
If \(q - 1 = 2^2 \cdot 13^1\), then \(q+1=54\) is a multiple of \(6\) and we have \(q=53\) and the answer is 'no'.
Since condition 1) does not yield a unique solution, it is not sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

Hi, how do we know which numbers have 6 factors? Is there a shortcut for this?