If n is a positive integer, is 24 a factor of n?

Question

(1) 24 is a factor of n^2.

(2) 24 is a factor of 2n.

sanjitscorps18 · Answer

24 = 2^3 x 3

Statement 1
If n = 24 the statement is true
If n = 12 the statement is true
--> Insufficient

Statement 2
If n = 24 then it is true
If n = 12 even then it is true
--> Insufficient

Since we have the example numbers common to both the statements
Both statements together are not sufficient

Option E

deekshaRaina · Answer

Could you pl help me understand why A is not the answer?

Basically, the question is asking if n/24 is an integer.

So, option A says that 24 is a factor of n*n, right. So that means that n*n/24 is an integer and 24 = 2*2*2*3 i.e all primes. So n*n must contain at least 1 of the primes therefore isn't n*n / 24 sufficient or correct?

sanjitscorps18 · Answer

I'll try to explain this one. So we have to look at the statement with the intention that 24 may or may not be the factor of n despite the fact that it appears outright that it should be...why isn't it?

24 = 2*2*2*3
So if you look at the factors we only need to multiply 24 by a single 2 and a single 3 to transform 24 into a perfect square. 
This perfect square is 2*2*2*3*  3*2   = 144. 
We can hence confirm that 144 is a perfect square (n^2) and is also divisible by 24. This is equivalent to saying n*2 is a multiple of 24, where n = 12.
However, n, in this case, is not divisible by 24. That is, 12 is not a multiple of 24
 Doesn't satisfy the expression asked

Now if I simply take n = 24, then we can easily see that 
n^2 = 24^2
But in this case, n is also a multiple of 24. 
 Hence this satisfies the expression asked

Since we are getting contrasting answers for the conditions specified, we cannot use Statement 1 as a solution.
Hope I was able to clarify your doubt.

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If n is a positive integer, is 24 a factor of n?

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If n is a positive integer, is 24 a factor of n?

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