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# If 100 < x^2 < 225, what is the greatest possible number of unique fac

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Math Expert
Joined: 02 Sep 2009
Posts: 52390
If 100 < x^2 < 225, what is the greatest possible number of unique fac  [#permalink]

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13 Feb 2017, 02:32
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Difficulty:

55% (hard)

Question Stats:

58% (01:31) correct 42% (01:31) wrong based on 107 sessions

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If 100 < x^2 < 225, what is the greatest possible number of unique factors of integer x?

A. 3
B. 4
C. 6
D. 12
E. 15

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Joined: 21 Mar 2016
Posts: 523
Re: If 100 < x^2 < 225, what is the greatest possible number of unique fac  [#permalink]

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13 Feb 2017, 03:07
since x is an integet,,,x^2 is a perfect square,, i.e x^2 = 121 or 144 or 169 or 196.
implies x is 11 or 12 or 13 or 14

calculating the number of factors for each,, 12 has highes number of factors i.e 6
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Re: If 100 < x^2 < 225, what is the greatest possible number of unique fac  [#permalink]

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13 Feb 2017, 08:43
Bunuel wrote:
If 100 < x^2 < 225, what is the greatest possible number of unique factors of integer x?

A. 3
B. 4
C. 6
D. 12
E. 15

$$100 < x^2 < 225$$

Or, + $$10$$ $$< x <$$ + $$15$$

So, value of x = $$+11 , -11 , +12 , -12 , + 13 , -13$$

Thus, there will be 6 possible unique values, answer will be (C) 6
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Re: If 100 < x^2 < 225, what is the greatest possible number of unique fac  [#permalink]

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22 Feb 2017, 19:59
1
Bunuel wrote:
If $$100 < x^2 < 225$$, what is the greatest possible number of unique factors of integer x?

A. 3
B. 4
C. 6
D. 12
E. 15

Official solution from Veritas Prep.

Since x is defined as an integer, you can quickly identify your possible values of x by looking at all the perfect squares between 100 and 225. Since 100 is 10-squared and 225 is 15-squared, your only options for x are 11, 12, 13, and 14.

From that list, you should recognize 11 and 13 as primes, meaning that they'll each have exactly two factors ("itself" and 1.) So you can eliminate 11 and 13 as options. From there you can factor the other two options.

14 has as factors 1 and 14, and 2 and 7 (remember with factors that it's helpful to think of them in pairs).

And 12 has as factors 1 and 12; 2 and 6; and 3 and 4. So the maximum number of unique factors of x is 6.
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Re: If 100 < x^2 < 225, what is the greatest possible number of unique fac &nbs [#permalink] 22 Feb 2017, 19:59
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