Divisibility/Multiples/Factors , Number Properties

[piyush2694]

How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

[piyush2694]

How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

OA: D

[himalaya savalia]

Suppose we want to write a number in its basic form, i.e. with help of prime numbers only.
$$Ex.\ 12\ =\ 2^2\ X\ 3$$

sentence 1 :
Now, if we take one factor of N which is greater than $$\sqrt{N}$$
another factor must be less than $$\sqrt{N}$$.
So, the first factor can not be the least prime factor.
Except when that number is prime number.
Ex. 11 = 11 * 1 where least prime factor 11 is greater than $$\sqrt{11}$$
So, N is prime number and have only one prime factor.
---> sufficient

sentence 2 :
This sentence directly implies that N is a prime number.
So, it must have only 1 prime factor.
---> sufficient

---> each statement alone is sufficient.
Ans. D

[Manasa3190]

1) If there is no prime number below the value of $$\sqrt{N}$$ which divides N then the number N is prime.
So N is a prime Number, which has only two factors 1 and itself
Only one prime number i.e., itself
Sufficient

2) A prime number has only two factors i.e., 1 and itself
So, only prime factor is N.
Sufficient
So, Option D is correct

[harsh8686]

Suppose we want to write a number in its basic form, i.e. with help of prime numbers only.
sentence 1 :
Now, if we take one factor of N which is greater than $$\sqrt{N}$$
another factor must be less than $$\sqrt{N}$$.
So, the first factor can not be the least prime factor.
Except when that number is prime number.
Ex. 11 = 11 * 1 where least prime factor 11 is greater than $$\sqrt{11}$$
So, N is prime number and have only one prime factor.
---> sufficient
Ans. D

I didn't get it. Can you please elaborate a bit. Perhaps by giving a reverse example.

[Manasa3190]

Example to illustrate 1)

Consider 239, whose $$\sqrt{239}$$ is in between 15 and 16, so we consider all prime numbers below 15 to check whether they are factors of 239
This is a shortest way to find whether a number is prime
Prime numbers less than 16 are 2,3,5,7,11,13
239 is not divisible by any of these prime numbers, so N is prime
When N is prime it has only two factors 1 and itself
So, number of prime factors is 1 Sufficient

eg 2:
Consider 15 which is not a prime number, $$\sqrt{15}$$ lies between 3 and 4
prime numbers below 4 are 2,3
15 is divisible by 3
So 15 is not a prime number

Basic rule is
If there is no prime number below the value of $$\sqrt{N}$$ which divides N then the number N is prime.