Number theory

[siddharth11.sharma]

If x and y are positive integers and 3x+5<x+11, is x a prime number?
1) The sum of x and y is even
2) The product of x and y is odd.

[jnelson0612]

If x and y are positive integers and 3x+5<x+11, is x a prime number?
1) The sum of x and y is even
2) The product of x and y is odd.

Siddharth, can you tell us the original source, please?

To solve, let's start by rephrasing the question.
1) Let's first unravel the equation: 3x + 5 < x + 11.
2) Subtract x from both sides and subtract 5 from both sides.
I'm left with 2x < 6.
3) Divide by 2 on both sides. I get x < 3. The question is whether x is a prime number, and there is only one prime number below 3, the number 2.
4) Thus, the question should be is x=2?

Statement 1 tells me x + y is even.
I am going to test numbers here. I have to put in numbers that fit the statement, and I will then use them to answer the question.
First, x could be 2 and y could be 4 as x+y=6, an even. That would allow us to answer our question whether x is prime YES.
BUT, x could also be 1 and y could be 3 as x+y=4, an even. Then we answer our question NO.
YES/NO means that statement 1 is insufficient!

Statement 2 tells me that the product of x * y is odd.
Well, the only way that I can get an odd product when multiplying two integers is if both integers are odd. Thus, x cannot be 2; it must be the odd integers 1 or below, none of which is prime. I answer the question NO for every single case. Remember, NO is a sufficient answer.
Thus, statement 2 is sufficient and my answer is B, 2 only is sufficient.

Please let us know if you need further clarification. :-)