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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.
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Since we have 1 variable (p) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.
Condition 1)
Since we have an infinite number of prime numbers, we don't have a unique value of p, and condition 1) is not sufficient.
Condition 2)
If p has a remainder 1 when it is divided by 3 or p=3k+1 for some integer k, then p^2+2 = (3k+1)^2+2 = 9k^2+6k+1+2 = 3(3k^2+2k+1) is a multiple and it is a prime number. We have 3k^2+2k+1=1, 3k^2+2k=0, k(3k+2)=0 and k=0 or k=-2/3. However, k is an integer so only k=0 works. Then p=3(0)+1 = 1. However, p = 1 is not a solution since 1 is not a prime number.
If p has a remainder 2 when it is divided by 3 or p=3k+2 for some integer k, then p^2+2 = (3k+2)^2+2 = 9k^2+12k+4+2 = 3(3k^2+4k+2) is a multiple and it is a prime number. Since we have 3k^2+4k+2=1, 3k^2+4k+1=0 or (3k+1)(k+1)=0 and we have k =-1 and k=-1/3. However, k must be an integer so then p=3(-1)+2 = -1. However, p = -1 is not a solution since -1 is negative.
Assume p has a remainder 0 when it is divided by 3.
If p=3, then p^2+2=11 is a prime number.
If p=9, then p^2+2=83 is a prime number.
Since condition 2) does not yield a unique solution, it is not sufficient.
Conditions 1) & 2)
p is a multiple of 3 from condition 2), and p is a prime number from condition 1). Then p = 3.
Since both conditions together yield a unique solution, it is sufficient.
Therefore, C is the answer.
Answer: C
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.
