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If x is a positive integer, then is x prime?
(1) 3x + 1 is prime
(2) 5x + 1 is prime
E
(1) 3x + 1 is prime
(2) 5x + 1 is prime
E
Given: x is a positive integerAbeNeedsAnswers wrote:If x is a positive integer, then is x prime?
(1) 3x + 1 is prime
(2) 5x + 1 is prime
E
To find out: is x prime?
Statement 1: 3x + 1 is prime
Since x is a positive integer, the minimum value of x = 1, thus the minimum value of 3x + 1 = 3*1 + 1 = 4, but '4' is non-prime, thus x = 1 is not a valid value.
Let's try with x = 2, then 3x = 1 = 3*2 + 1 = 7, a prime number. So, if x = 2, then x is prime.
We know that all the prime numbers except 2 are odd, thus, 3x + 1 = Odd.
3x + 1 = Odd
3x = Odd - 1 = Odd - Odd = Even
x = Even / 3 = Even / Odd = Even (since x is a positve integer);
Note that if an even integer is completely divisible by an odd integer, the quotient is even.
So, we have x = Even
If x = 2, the answer is Yes; however, if x ≠2, the answer is No. Insufficient.
Statement 2: 5x + 1 is prime
At x = 2, we see that 5x + 1 = 5*2 + 1 = 11, a prime number, so if x = 2, the answer is yes.
As discussed in Statement 1, we know that all the prime numbers except 2 are odd, thus, 5x + 1 = Odd.
5x + 1 = Odd
5x = Odd - 1 = Odd - Odd = Even
x = Even / 5 = Even / Odd = Even (since x is a positve integer);
So, we have x = Even
If x = 2, the answer is Yes; however, if x ≠2, the answer is No. Insufficient.
Statement 1 & 2:
x = 2, implying x is prime, and x = other even integers are applicable in both the statements, thus there is no unique answer. Insufficient.
The correct answer: E
Hope this helps!
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-Jay
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