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gander123
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Hi guys,
I recently reviewed the following question but still haven't quite understood the answer explanation:
"If x is a positiv integer, is √x an integer?"
(1) √4*x is an integer.
(2) √3*x is not an integer.
Answer: Only statement (1) is sufficient --> A.
My question:
First of all, I fully understand, why statement (2) is not sufficient alone. Therefore, I will leave that part out here.
However, I am really struggling with the argumentation for statement (1):
"It is given that √4*x = n, or 4*x = n², for some positive integer n. Since 4*x is the square of an integer, it follows that in the prime factorization of 4*x, each distinct prime factor is repeated an even number of times. (How do you know this??). Therefore, the same must be true for the prime factorization of x, since the prime factorization of x only differs from the prime factorization of 4*x by two factors of 2, and hence by an even number of factors of 2;SUFFICIENT"
Can anyone out there help me out with an applied answer ?
My assumption:
4*x is a squared integer, and a squared integer is always an integer. Due to the theory (presented in GMAT Math Review) of integers, one knows that every integer can be expressed as the product of (an odd or even number) of prime factors. Since we have here two times a number of odd or even prime factors (for n²) the number of prime factors has to be even for the whole expression 4*x anyway?!
Am I getting it the right way ?
I'd appreciate your detailed and structured responses.
Kind regards,
Tobi
I recently reviewed the following question but still haven't quite understood the answer explanation:
"If x is a positiv integer, is √x an integer?"
(1) √4*x is an integer.
(2) √3*x is not an integer.
Answer: Only statement (1) is sufficient --> A.
My question:
First of all, I fully understand, why statement (2) is not sufficient alone. Therefore, I will leave that part out here.
However, I am really struggling with the argumentation for statement (1):
"It is given that √4*x = n, or 4*x = n², for some positive integer n. Since 4*x is the square of an integer, it follows that in the prime factorization of 4*x, each distinct prime factor is repeated an even number of times. (How do you know this??). Therefore, the same must be true for the prime factorization of x, since the prime factorization of x only differs from the prime factorization of 4*x by two factors of 2, and hence by an even number of factors of 2;SUFFICIENT"
Can anyone out there help me out with an applied answer ?
My assumption:
4*x is a squared integer, and a squared integer is always an integer. Due to the theory (presented in GMAT Math Review) of integers, one knows that every integer can be expressed as the product of (an odd or even number) of prime factors. Since we have here two times a number of odd or even prime factors (for n²) the number of prime factors has to be even for the whole expression 4*x anyway?!
Am I getting it the right way ?
I'd appreciate your detailed and structured responses.
Kind regards,
Tobi
