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If g is a the square of an integer, then what is the value

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If g is a the square of an integer, then what is the value [#permalink] New post 20 Sep 2006, 14:53
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If g is a the square of an integer, then what is the value of g?

(1) Fewer than half of the factors of g are less than 12.
(2) Fewer than half of the factors of g are greater than 12.
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 [#permalink] New post 21 Sep 2006, 06:28
I think it "might" be C.

Statement 1 by itself is not sufficient. The number could 169 or 289 or the square of any prime number greater than 12 and the condition would hold.

Statement 2 by itself is not sufficient. The number could be 16 or 25 and the condition would hold.

Now the question is can we solve this with both together.

By picking different numbers, both these conditions cannot be satisfied simultaneously for any other number except 144. In fact if we pick any square of an integer, it always has an equal number of factors less than its root as it has greater than its root. For example 16 has two factors less than its root 4, i.e. 1 and 2 and 2 factors greater than its root i.e. 8 and 16.

I guess the proof of this would involve the rule that the greatest prime factor of a number is always less than its root. Based on this (and maybe permutations) we should be able to derive the rule on the number of factors

Hence i would say that the number is 144 and solution is C.
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 [#permalink] New post 25 Sep 2006, 03:42
Hey Kevin, can you share the OA for this one? Thanks!
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Re: PS: The value of g [#permalink] New post 25 Sep 2006, 04:30
kevincan wrote:
If g is a the square of an integer, then what is the value of g?

(1) Fewer than half of the factors of g are less than 12.
(2) Fewer than half of the factors of g are greater than 12.


(1) g could be 13^2, 17^2,... NOT SUFF
(2) g could be 5^2, 7^2,... NOT SUFF

(1) and (2) 12 is a factor of g. As there is a one-to-one correspondence between factors of g that are less than sqrt(g) and those that are greater than sqrt(g), there are two possibilities:

(A) there is an even number of factors, half less than sqrt(g) and half above sqrt(g)

(B) there is an odd number of factors. In this case less than half of the factors are are less than sqrt(g) and less than half are greater than sqrt(g). The remaining factor is equal to sqrt(g).


Clearly A is not the case, so it must be that 12=sqrt(g) SUFF ANS:C
Re: PS: The value of g   [#permalink] 25 Sep 2006, 04:30
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If g is a the square of an integer, then what is the value

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