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Divisibility/Multiples/Factors , Number Properties

Problem Solving — algebra and arithmetic
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by piyush2694 » Thu Dec 20, 2018 4:01 am
How many prime factors does N have?

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

(2) N has 2 positive factors.
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Source: — Quantitative Reasoning |
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Numbers, Prime Numbers

by piyush2694 » Thu Dec 20, 2018 9:16 am
piyush2694 wrote: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
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by himalaya savalia » Sun Dec 23, 2018 11:33 am
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
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Divisibility/Multiples/Factors , Number Properties

by Manasa3190 » Sat Dec 29, 2018 7:05 pm
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
Last edited by Manasa3190 on Sat Dec 29, 2018 9:39 pm, edited 1 time in total.
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by harsh8686 » Sat Dec 29, 2018 9:19 pm
himalaya savalia wrote: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.
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by Manasa3190 » Sat Dec 29, 2018 9:50 pm
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.
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