This is an excellent question on the properties of Primes. I’d also use the Units digit concept to solve the question since there is a multiple of 5 mentioned in one of the statements.
‘x’ is a positive integer. This means that x can take any value from the set {1,2,3,4,5……}.
From statement I, we know that 3x + 1 is prime. Since x is a positive integer, we can say that the smallest value we can take for 3x is 3, which is to say that the smallest value for x is 1. Therefore, 3x + 1 cannot be equal to 2 or 3.
If 3x + 1 = 5, 3x = 4. x is not an integer here. If 3x + 1 = 7, 3x = 6 which gives us x = 2. This value of x answers the main question with a YES since 2 is prime.
Any prime number greater than 3 can be written as 6k – 1 or 6k + 1 where k is a positive integer. A careful analysis will tell us that when we equate 3x + 1 to 6k + 1, we get x to be an integer. That’s why we got x = 2 in the example above.
The next two prime numbers that can be written in the form of 6k + 1 are 13 and 19. If 3x + 1 = 13, x = 4; if 3x + 1 = 19, x = 6. 4 and 6 are not prime numbers and hence answer the main question with a NO.
Statement I is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.
From statement II alone, 5x + 1 is a prime. Again, the smallest prime of this form can be 11. Clearly we are dealing with odd primes. Therefore, 5x has to be even. In other words, 5x is a multiple of 5 with units digit ZERO and so 5x + 1 is a prime with units digit 1. This means that x can be 2 or 6 or 8. We again face a YES NO situation.
Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.
Combining the data given in the two statements, we have x = 2 and x = 6 satisfying both statements at the same time. The combination of statements is insufficient as well.
The correct answer option is E.
Hope that helps!
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