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Let a, b, and c be three integers, and let a be a perfect square. If a

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Let a, b, and c be three integers, and let a be a perfect square. If a  [#permalink]

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21 May 2017, 05:20
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Let a, b, and c be three integers, and let a be a perfect square. If a/b = b/c, then which one of the following statements must be true?

(A) c must be an even number
(B) c must be an odd number
(C) c must be a perfect square
(D) c must not be a perfect square
(E) c must be a prime number

Source: Nova GMAT

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Re: Let a, b, and c be three integers, and let a be a perfect square. If a  [#permalink]

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21 May 2017, 19:02
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Given: $$\frac{a}{b}$$=$$\frac{b}{c}$$
Thus, $$b^2$$= ac
It is given that a is a perfect square
From the above equation, It is concluded that
b is also a perfect square
Therefore, C must be a perfect square

$$Answer: C.$$

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Re: Let a, b, and c be three integers, and let a be a perfect square. If a  [#permalink]

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10 Sep 2018, 00:29
varun4s wrote:
Given: $$\frac{a}{b}$$=$$\frac{b}{c}$$
Thus, $$b^2$$= ac
It is given that a is a perfect square
From the above equation, It is concluded that
b is also a perfect square
Therefore, C must be a perfect square

$$Answer: C.$$

Kudos please if you like my explanation!

Hi!

I don't quite get how C is obviously a perfect square too, could someone elaborate on that?

Thank you!
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Re: Let a, b, and c be three integers, and let a be a perfect square. If a  [#permalink]

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10 Sep 2018, 01:01
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Let a, b, and c be three integers, and let a be a perfect square. If a/b = b/c, then which one of the following statements must be true?

(A) c must be an even number
(B) c must be an odd number
(C) c must be a perfect square
(D) c must not be a perfect square
(E) c must be a prime number

Source: Nova GMAT

a, b and c are integers.

$$a/b = b/c$$

$$a*c = b^2$$

The right hand side is a perfect square (since it is the square of an integer, say b = 6 so b^2 = 6^2 = 36 = 2^2 * 3^2). All prime factors of a perfect square are in pairs.
Then the left hand side must be a perfect square too i.e. have all prime factors in pairs.
Now we are given that a is a perfect square. This implies all prime factors in a are in pairs. (Say a = 4 = 2^2)
Then all leftover prime factors must be in pairs too (i.e. 3^2). So c must be a perfect square too.

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Re: Let a, b, and c be three integers, and let a be a perfect square. If a  [#permalink]

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10 Sep 2018, 07:39
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Let a, b, and c be three integers, and let a be a perfect square. If a/b = b/c, then which one of the following statements must be true?

(A) c must be an even number
(B) c must be an odd number
(C) c must be a perfect square
(D) c must not be a perfect square
(E) c must be a prime number

Source: Nova GMAT

$$a = k^2$$

So, $$\frac{a}{b} = \frac{b}{c}$$

Or, $$a = \frac{b^2}{c}$$

Hence, $$k^2 = \frac{b^2}{c}$$

Or, $$c = \frac{b^2}{k^2}$$

Or, $$c = (\frac{b}{k})^2$$, Answer must be (C) c must be a perfect square
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Re: Let a, b, and c be three integers, and let a be a perfect square. If a &nbs [#permalink] 10 Sep 2018, 07:39
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