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# If 220n is the square of a positive integer, what is the smallest poss

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Math Expert
Joined: 02 Sep 2009
Posts: 61283
If 220n is the square of a positive integer, what is the smallest poss  [#permalink]

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20 Jan 2020, 01:43
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83% (01:06) correct 17% (01:46) wrong based on 18 sessions

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If 220n is the square of a positive integer, what is the smallest possible value of integer n?

(A) 15
(B) 25
(C) 35
(D) 45
(E) 55

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Joined: 20 Jul 2017
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Location: India
Concentration: Entrepreneurship, Marketing
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Re: If 220n is the square of a positive integer, what is the smallest poss  [#permalink]

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20 Jan 2020, 03:43
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Bunuel wrote:
If 220n is the square of a positive integer, what is the smallest possible value of integer n?

(A) 15
(B) 25
(C) 35
(D) 45
(E) 55

$$220n = 4*55*n = 2^2*5^1*11^1*n$$

Clearly, the least value of 'n' for which $$2^2*5^1*11^1*n$$ will be a perfect square = $$5^1*11^1 = 55$$

Option E
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Re: If 220n is the square of a positive integer, what is the smallest poss  [#permalink]

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20 Jan 2020, 03:52
For a number to be a perfect square of a positive integer, it is necessary for the power of each of its prime factor (unique) to be an even number.

Prime factorizing 220:

220 = $$2^{2}$$ x 5 x 11
220n = $$2^{2}$$ x 5 x 11 x n

The base 2 already has an even power. If 220n is a perfect square of a positive integer, the powers of 5 and 11 must also be even numbers.

The smallest value of n for which this is possible is n = 5 x 11 = 55

In this case:

220n
= $$2^{2}$$ x 5 x 11 x (5 x 11)
= $$2^2$$ x $$5^2$$ x $$11^2$$

which is a perfect square

Hope this helps.
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Re: If 220n is the square of a positive integer, what is the smallest poss   [#permalink] 20 Jan 2020, 03:52
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