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Is the positive integer n the square of an integer?

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Is the positive integer n the square of an integer?  [#permalink]

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New post 19 Feb 2019, 10:50
2
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

56% (02:03) correct 44% (01:27) wrong based on 48 sessions

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Is the positive integer n the square of an integer?


1) 4n is the square of an integer

2) n^3 is the square of an integer
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Re: Is the positive integer n the square of an integer?  [#permalink]

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New post 20 Feb 2019, 04:46
2
This is a bit abstract, it's best to use the Alternative approach and plug in numbers:
1) the condition holds and answer is "yes" for 1 and 4, "yes" for 4 and 16, yes for 16 and 64... and so on! sufficient!
2)this condition holds, and the answer is "yes" for 1 & 1, 4 and 64, for whom the answer is "yes" - sufficient!
D.
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Re: Is the positive integer n the square of an integer?  [#permalink]

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New post 20 Feb 2019, 12:28
1
If a number is the square of an integer, then in the prime factorization of that number, all of the exponents will be even. For example, (3^8)(5^6) is the square of an integer (it is the square of (3^4)(5^3) ) whereas (3^7)(5^6) is not.

Statement 1 tells us that in the prime factorization of (2^2)(n), all of the exponents are even. The prime factorization of n is identical to that of (2^2)(n) except that in n, the exponent on the '2' is smaller by two, and subtracting two doesn't change an even number to an odd number. So if the exponents in (2^2)(n) are even, so are the exponents in n, and Statement 1 is sufficient.

Statement 2 tells us that in the prime factorization of n^3, all of the exponents are even. The exponents in the prime factorization n^3 are all 3 times as big as the exponents in the prime factorization of n. Multiplying by 3 doesn't change evenness or oddness, so if all the exponents in n^3 are even, so are the exponents in n, and Statement 2 is also sufficient.

The answer is D.
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Re: Is the positive integer n the square of an integer?  [#permalink]

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New post 25 Mar 2019, 04:36
Pdirienzo wrote:
Is the positive integer n the square of an integer?


1) 4n is the square of an integer

2) n^3 is the square of an integer


prime factorization is key. too hard problemn

4n =a^2
prime factorization a^2= x^2k=x^2 or x^4 or x^6...
4n=x^2k
x^2k= even
x=even
x=(2y)^2k
n must be a square

go to 2 nd condition
n^3=a^2
a^2 = x^2a*y^2b, with x,y are prime
n^3=x^2a*y^2b , this mean a and b must be multiple of 3
this mean 2a and 2b are multiple of 6
n^3=x^6k*y^6m
this mean
n=x^2k*y^2m
this mean n is a squre of integer.

too hard.
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Re: Is the positive integer n the square of an integer?   [#permalink] 25 Mar 2019, 04:36
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Is the positive integer n the square of an integer?

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