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If k is a positive integer, is k the square of an integer?

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If k is a positive integer, is k the square of an integer?  [#permalink]

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New post 31 Jul 2017, 10:21
2
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A
B
C
D
E

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  45% (medium)

Question Stats:

70% (01:11) correct 30% (01:26) wrong based on 40 sessions

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If k is a positive integer, is k the square of an integer?
(1) k+1 is divisible by only two different numbers.
(2) k is divisible by only nine different numbers.

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Re: If k is a positive integer, is k the square of an integer?  [#permalink]

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New post 31 Jul 2017, 10:55
Should be B

Statement 1: k+1 is prime number, if k is 4 then it is a square of an integer, however if k is 12 it isn't a square of an integer. Insufficient.

Statement 2: k is z^8 or lets say (x^2)*(y^2)=k which is always a square of an integer if k is a positive integer.

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Re: If k is a positive integer, is k the square of an integer?  [#permalink]

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New post 31 Jul 2017, 22:39
I would like to add a perspective to the second statement.

S2 says that k is divisible by only 9 different numbers. This means that k has 9 unique factors.

Now, note that factors of an integer ALWAYS occur in pairs. For example : 8 = (1,2,4,8). By pairs, I mean that the product of first and last factor will give you the integer ( 1*8=8 ). The second and the second last factor will give you the integer ( 2*4=8 ). Hence, the pairs are 1,8 and 2,4.

Now, lets take an integer with odd number of factors. For example 4 = (1,2,4). Here the pairs are 1,4. Wondering what 2 will pair with ? Yes, it will pair with 2 itself !! So you have a 4 as a SQUARE of an integer !!

Hence, you can conclude that any integer with an odd number of factors will be the square of its middle factor. Another example is 9 = (1,3,9). 9 is the square of its middle factor !!

So we can conclude that k, which has 9 factors (odd) will be a square of its middle factor. Therefore S2 is sufficient.

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Re: If k is a positive integer, is k the square of an integer? &nbs [#permalink] 31 Jul 2017, 22:39
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