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# For positive integer x, if x^2 has 4 digits, which of the following

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Manager
Joined: 10 Feb 2011
Posts: 110
For positive integer x, if x^2 has 4 digits, which of the following [#permalink]

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17 Feb 2011, 15:06
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Difficulty:

65% (hard)

Question Stats:

55% (01:29) correct 45% (01:35) wrong based on 112 sessions

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For positive integer x, if x^2 has 4 digits, which of the following must be true?

I. x must be a 2-digit integer
II. 2x must be a 3-digit integer
III. 3x must be a 3-digit integer

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
Math Expert
Joined: 02 Sep 2009
Posts: 47047
Re: For positive integer x, if x^2 has 4 digits, which of the following [#permalink]

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17 Feb 2011, 16:15
1
banksy wrote:
186. For positive integer x, if x^2 has 4 digits, which of the following must be true?
I. x must be a 2-digit integer
II. 2x must be a 3-digit integer
III. 3x must be a 3-digit integer
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

As $$x$$ is an integer and $$x^2$$ is a 4-digit perfect square then $$32\leq{x}\leq{99}$$ (32^2=1024 is the first 4-digit perfect square and 100^2=10,000 is already a 5-digit perfect square).

Hence only I must be true (if $$x=32$$ then $$2x=64$$ and $$3x=96$$ so not the 3-digit integers).

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Re: For positive integer x, if x^2 has 4 digits, which of the following [#permalink]

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31 Mar 2017, 12:34
banksy wrote:
For positive integer x, if x^2 has 4 digits, which of the following must be true?

I. x must be a 2-digit integer
II. 2x must be a 3-digit integer
III. 3x must be a 3-digit integer

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Since x is a positive integer and x^2 has 4 digits, the minimum value of x is 32 since 32^2 = 1,024 (and 31^2 = 961) and the maximum value of x is 99 since 99^2 is 9,801 (and 100^2 = 10,000).

Thus, we see that x MUST BE a two-digit integer; however, since 2 x 32 = 64 and 3 x 32 = 96, neither 2x nor 3x has to be a three-digit integer.

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Re: For positive integer x, if x^2 has 4 digits, which of the following [#permalink]

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14 Aug 2017, 02:58
banksy wrote:
For positive integer x, if x^2 has 4 digits, which of the following must be true?

I. x must be a 2-digit integer
II. 2x must be a 3-digit integer
III. 3x must be a 3-digit integer

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

X^2 has 4 digits. So,1000< x^2 <10000
31< x<100
32<=x<=99 , x must be a 2 digit integer.

64 <=2x<=198, x can be a 2 digit/3digit integer

96 <=3x<=297, x can be a 2 digit/3digit integer

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Re: For positive integer x, if x^2 has 4 digits, which of the following   [#permalink] 14 Aug 2017, 02:58
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