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Is x a prime number, given that x is a positive integer?

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New post 18 Nov 2018, 10:30
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Is \(x\) a prime number, given that \(x\) is a positive integer?

(1) \(x^4 > 3000\)
(2) \(x^4 < 10000\)

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Is x a prime number, given that x is a positive integer?  [#permalink]

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New post Updated on: 18 Nov 2018, 13:53
Misread and thought one condition is to the power of 3.

Originally posted by philipssonicare on 18 Nov 2018, 13:29.
Last edited by philipssonicare on 18 Nov 2018, 13:53, edited 1 time in total.
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Is x a prime number, given that x is a positive integer?  [#permalink]

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New post 18 Nov 2018, 13:43
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Found the question online while looking at theory on squares of prime numbers. Here is the explanation:

Stmt 1: There are infinite possible values for which \(x^4\) is greater than 3000, hence, cannot determine if x is prime or not. For e.g. \(10^4\) and \(11^4\) both are greater than 3000.

Insufficient. Eliminate A,D

Stmt 2: There are 9 positive integers satisfying this condition, i.e. 1 through 9, \(1^4\) = 1, \(9^4\) = 6561, which are less than 10,000. Hence Insufficient. Eliminate B.

Combined: This gives us the condition: 3,000 < \(x^4\) < 10,000
Only 2 integers satisfy this condition, namely 8 and 9. But since both integers are NOT prime, we can say that x is surely NOT prime. Hence sufficient!

Answer is C

Hope that helps!
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Re: Is x a prime number, given that x is a positive integer?  [#permalink]

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New post 21 Dec 2018, 10:40
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Is x a prime number, given that x is a positive integer?

(1) x^4>3000
(2) x^4<10000

Ans: From 1: Let x = 7 => X^4 = (49 x 49) ~= (50x 50) = 2500 <3000
if x = 8 => X^4 = (64 x 64) ~= (60x60) = 3600>3000
so x >= 8.....no unique value so Not sufficient

From 2: x^4 <10000 => X <10....no unique value so Not sufficient
From 1 and 2 : 3000<x<10000
=> x= 8, 9 ....both are not prime numbers
Hence, x is not a prime no (C) !!
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Re: Is x a prime number, given that x is a positive integer?  [#permalink]

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New post 22 Dec 2018, 13:10
nkin wrote:
Is \(x\) a prime number, given that \(x\) is a positive integer?

(1) \(x^4 > 3000\)
(2) \(x^4 < 10000\)


Here's how I solved the problem with approximation.

Statement 1: x^4 > 3000
Clearly insufficient since there can be many primes and non-primes if an "upper limit" is not given.

Statement 2: x^4 < 10000
That is, x^4 < 10^4
Therefore, x<10
Clearly insufficient since x can be any number from 2 to 9 - and therefore may be prime or non prime


Statement 1 & 2 together:
1) gives us x^4 > 3000
Let's simplify by taking square root.
50^2 = 2500
60^2 = 3600
Difference is 1100 which is close to the average of the 2 nos.
Checking for 55^2, we get 3025, which is greater than 3000. So definitely 54^2 is less than 3000.
Hence, surely x^2 > 54.
Let's simplify further by taking square root.
7^2 = 49 and 8^2 = 64. Pick 7^2 since it's lesser than 54.
Hence, surely x > 7 ................(3)

From equation (3) and statement (2),
we have x is greater than 7, and x is less than 10.
That is, 7 < x < 10.
So, x is either 8 or 9 (since x is an integer.)
Therefore, x is definitely not a prime number.
Hence, sufficient.

:)
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Re: Is x a prime number, given that x is a positive integer?   [#permalink] 22 Dec 2018, 13:10
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