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# For what smallest value of x, 350*x will be square of an Integer? A)

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For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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25 Sep 2018, 08:36
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85% (00:59) correct 15% (01:02) wrong based on 33 sessions

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For what smallest value of x, 350*x will be square of an Integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: www.GMATinsight.com

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For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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25 Sep 2018, 08:58
GMATinsight wrote:
For what smallest POSITIVE INTEGER value of x will 350*x be the square of an integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: http://www.GMATinsight.com

$$x \geqslant 1\,\,\operatorname{int}$$

$$? = \min \,x$$

Important: any perfect square greater than 1 is characterized as the product of distinct primes, each prime with an even power.

$$\left. \begin{gathered} 350x = 2 \cdot {5^2} \cdot 7 \cdot x \hfill \\ 350x\,\, = {\text{prime}}{{\text{s}}^{\,{\text{even}}}}\,\, \hfill \\ \end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,? = 2 \cdot 7 = 14$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: note that 350*0 = 0 is the square of an integer.
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Re: For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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25 Sep 2018, 13:07
GMATinsight wrote:
For what smallest value of x, 350*x will be square of an Integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: http://www.GMATinsight.com

In order to make 350*x a perfect square we have to find out the dna of this number.

$$350x = 5*5*7*2 *x = 5^2 * 7 * 2 x.$$

Since 350x has to be perfect square we need additional one 7 and one 2. It means x has to be 14.

$$350*14 = 5^2*7^2*2^2$$

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Re: For what smallest value of x, 350*x will be square of an Integer? A)  [#permalink]

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26 Sep 2018, 01:35
1
1
GMATinsight wrote:
For what smallest value of x, 350*x will be square of an Integer?

A) 2
B) 5
C) 7
D) 10
E) 14

Source: http://www.GMATinsight.com

CONCEPT: A number is a Perfect Square when exponents of all distinct prime factors of the number are EVEN integers

Key to answer such questions is PRIME FACTORIZATION of the number

$$350*x = 35*10*x = 2*5^2*7*x$$

In order to make exponents of 2 and 7 Even integers we need x to contribute one 2 and one 7 i.e.e $$x = 2*7=14$$ for least value of x

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Re: For what smallest value of x, 350*x will be square of an Integer? A)   [#permalink] 26 Sep 2018, 01:35
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