What number only has 2 divisors?

$N=11$ is also possible.

All the positive divisors of a positive integer $N$ includes at least $N$ and $1$ as its divisors. Thus,

$N+1+\text{Sum of the other divisors}=12$ ' $\text{Sum of the other divisors}=11-N$ $\text{(Assuming N to be greater than 1, as N only has 1 divisor i.e 1, and clearly 1 is not equal to 12)}$

Now, $N$ has to be less than or equal to 11 and greater than 1 as per our assumption, otherwise, the sum of other divisors will result in a negative number and positive numbers don't add up to something negative. It could be equal to 11 as 0 would mean there are no other divisors. We can now remove 2, 3, 4, 5, 7 as their divisors add less than 12, and 8, 9, 10 as its sum of divisors is greater than 12. Thus only 6, and 11 are possible.