Data Sufficiency

If n is a positive integer, is n divisible by 2?

A)7n-8 is divisible by 20.
B)3n^2+2n+5 is a prime number.

OA: D

Hello NandishSS.

I will try to solve your question.

We know that n is a positive integer and we want to know if n is divisible by 2.

First Statement

A)7n-8 is divisible by 20.

This statement tells us that $$7n-8=20\cdot k\ \ \ ,\ \ \ k\in\mathbb{Z}$$ $$\Rightarrow\ \ 7n=20\cdot k\ +8\ \ ,\ \ \ k\in\mathbb{Z}$$ $$\Rightarrow\ \ 7n=2\left(10\cdot k\ +4\right)\ \ ,\ \ \ k\in\mathbb{Z}$$ The last equation says that 7b is divisible by 2 and since 7 is not divisible by 2, we conclude that n is divisible by 2.

Hence, this statement is SUFFICIENT.

Second Statement

B)3n^2+2n+5 is a prime number.

Here we have to see the following:

- Since n is a positive integer then n>0 (n >= 1)
- If n>0 then 3n^2+2n+5>= 10.
- Since 3n^2+2n+5 is a prime number and it is greater or equal than 10, then 3n^2+2n+5 must be an odd number.

So, we have the following: $$3n^2+2n+5=odd\ \ \ \Rightarrow\ \ 3n^2+2n=odd-5\ \ \ \Rightarrow\ \ \ 3n^2+2n=even.$$ $$\Rightarrow\ \ 3n^2=even\ -\ 2n\ \ \ \ \Rightarrow\ \ \ 3n^2=even\ -even\ \ \ \Rightarrow\ \ \ 3n^2=even.$$ Since 3 is odd, then we have to n^2 must be even. This only happens when n is even.

Therefore, n is even and this implies that n is divisible by 2.

So, this statement is also SUFFICIENT.

In conclusion, each statement is SUFFICIENT, so the correct answer is the option D.

I hope it helps.

Regards.