If n is a positive integer and is not the square of any integer, is it a prime number?
(1) Among the factors of n, only n is greater than √n
(2) among the factors of n, only 1 is less than √n
OA D
Totally stumped!! Can someone explain?
Prime numbers
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if n=15, root(15) = 3. something..
5> root(15) which is a factor.
A) means its prime. any number which is not prime and not square, will have atleast 1 factor greater than root(n)
B) same as A...
IMO D
5> root(15) which is a factor.
A) means its prime. any number which is not prime and not square, will have atleast 1 factor greater than root(n)
B) same as A...
IMO D
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If you take a positive integer x which is not a perfect square, then if x is *not* prime, it must be possible to write x as a product of two smaller integers:tdkk123 wrote:If n is a positive integer and is not the square of any integer, is it a prime number?
(1) Among the factors of n, only n is greater than √n
(2) among the factors of n, only 1 is less than √n
OA D
Totally stumped!! Can someone explain?
x = a*b
If x is not a perfect square, then a and b must be *different* (and if a and b are both less than 24, neither of them is equal to 1). So, for example, if x were 24, we could write x as a product of two smaller integers in several ways (2*12, 3*8, 4*6). Now, if x = a*b, then one of the numbers a or b must be smaller than √x, and the other must be larger than √x. They can't both be smaller than √x, because then their product would be smaller than √x*√x = x, and they similarly can't both be larger than √x, because then their product would be greater than x. In the specific example of 24, you can see that in each of the three products (2*12, 3*8, 4*6) one of the factors is less than √24 ≈ 5, and one is greater.
So if x is not a prime, and not the square of a prime, x must have at least one factor which is somewhere strictly between 1 and √x, and another which is strictly between √x and x. That makes the answer D here.
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The statements say, Among the factors of, say, 24, only 24 is greater than sqrt(24).
However, there is more than one factor less than sqrt(24). (1*24)(2*12)(3*8)(4*6). Similarly, there is more than one factor greater than sqrt(24)
However, there is more than one factor less than sqrt(24). (1*24)(2*12)(3*8)(4*6). Similarly, there is more than one factor greater than sqrt(24)
In case of confusion solve it using examples...
The question asked is if n is a prime number...
option a : 1st take an example which is not a prime number...
say 26(even number)...which has factors (2*13) or (1*26)...
or 27(odd number)...which has factors (3*9) or (1*27)...
root of 26 ~ 5.something...as seen both 13 and 26 are greater than root n...
similarly even for 27...as seen both 9 and 27 are greater than root n...
if n is a prime number like 13...factors are (1*13)...
therefore only 13 is greater than root 13 ~ 9.something
therefore n has to be a prime number to satisfy the conditions given in a...
option b : as seen from the above example 26(has factors 1 and 2 less then root of 26) and 27 (has factors 1 and 3 less than root of 27)...prime number 13 has only 1 less than root 13...therefore in both the cases n has to be a prime number...
I hope this could be of some help...
The question asked is if n is a prime number...
option a : 1st take an example which is not a prime number...
say 26(even number)...which has factors (2*13) or (1*26)...
or 27(odd number)...which has factors (3*9) or (1*27)...
root of 26 ~ 5.something...as seen both 13 and 26 are greater than root n...
similarly even for 27...as seen both 9 and 27 are greater than root n...
if n is a prime number like 13...factors are (1*13)...
therefore only 13 is greater than root 13 ~ 9.something
therefore n has to be a prime number to satisfy the conditions given in a...
option b : as seen from the above example 26(has factors 1 and 2 less then root of 26) and 27 (has factors 1 and 3 less than root of 27)...prime number 13 has only 1 less than root 13...therefore in both the cases n has to be a prime number...
I hope this could be of some help...