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# If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values

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If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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07 Feb 2016, 10:55
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If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values of p will make N a perfect square?

A. 0
B. 2
C. 3
D. 5
E. 7
[Reveal] Spoiler: OA

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Last edited by Bunuel on 11 Jul 2017, 08:50, edited 1 time in total.
Edited the OA.

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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07 Feb 2016, 12:57
Bunuel wrote:
If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values of p will make N a perfect square?

A. 0
B. 2
C. 3
D. 5
E. 7

I think the answer should be C.
Thoose are P-Values that satisfy given restriction: -2, 0, 2 (note, a fraction can be also a perfect square) --> 16*100, 16*1, 16/100
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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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10 Jul 2017, 21:47
Can someone explain the definition of a perfect square to me?

I thought that in order to be a perfect square, a number had to be the result of an integer times itself, but is the only constraint that the root be rational?

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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10 Jul 2017, 21:59
B.

-2 & 0.

2 will make it 4^2*10^-2.. which isn't a perfect square ? What am I missing ?

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If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 07:38
Quote:
If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values of p will make N a perfect square?

A. 0
B. 2
C. 3
D. 5
E. 7

SOUMYAJIT_ wrote:
B.

-2 & 0.

2 will make it 4^2*10^-2.. which isn't a perfect square ? What am I missing ?

Bolded part - .16 can be a perfect square if the latter's definition is "a rational number multiplied by itself." ... Which seems to be the issue.

If -p = 2, we get p = -2, and 16 x 10$$^{-2}$$ = .16 --> .4 * .4

If -p = (-2), we have p = 2. 16 * 10$$^2$$ = 1600 = 40 * 40

My solution: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values of p will make N a perfect square?

1. -p has to be -3, -2, -1, 0, 1, 2, or 3

2. Check the integers

16 x 10$$^{-(-3)}$$ =16 * 1000 . . . 1000 is not a perfect square. Incorrect.

16 x 10$$^{-(-2)}$$ = 1600. 40*40 = 1600. Correct

16 x 10$$^{-(-1)}$$ = 16 * 10. 10 is not a perfect square. Incorrect

16 x $$10^{-(-1)(0)}$$ = $$10^0$$ = 16 * 1 = 16. 16 = 4*4. Correct.

16 * $$10^{-1}$$ = 1.6, or 1$$\frac{6}{10}$$, or $$\frac{16}{10}$$, whose denominator is not a perfect square. Incorrect.

16 * 10$$^{-2}$$ = .16 = (.4)$$^2$$. Also, .16 = $$\frac{16}{100}$$. Numerator and denominator are perfect squares. Correct.

16 * 10$$^{-3}$$ = .016 = $$\frac{16}{1000}$$. 1000 is not a perfect square. Incorrect.

3. -p = -2, 0, and 2. There are three integer values of -p that will make N a perfect square.

brianne5 , I have the same question as you do, though I answered the question as if the definition of perfect square were expanded beyond integers. I wonder if GMAT adheres to the definition of perfect square as the product of a rational number multiplied by itself? Even in the Math Theory section on the forum, a perfect square is defined as "A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is an perfect square." https://gmatclub.com/forum/math-number-theory-88376.html

Edited to use -p correctly. Answer unchanged.

Last edited by genxer123 on 11 Jul 2017, 08:59, edited 2 times in total.

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:10
genxer123 wrote:
Quote:
If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values of p will make N a perfect square?

A. 0
B. 2
C. 3
D. 5
E. 7

SOUMYAJIT_ wrote:
B.

-2 & 0.

2 will make it 4^2*10^-2.. which isn't a perfect square ? What am I missing ?

Not sure how you got bolded part. If p = 2, we get 10$$^2$$. 16 * 10$$^2$$ = 1600 = 40 * 40

My solution: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values of p will make N a perfect square?

1. p has to be -3, -2, -1, 0, 1, 2, or 3

2. Check the integers

16 x 10$$^{-3}$$ = .016 = $$\frac{16}{1000}$$. 1000 is not a perfect square. Incorrect.

16 x 10$$^{-2}$$ = .16 = (.4)$$^2$$. Also, .16 = $$\frac{16}{100}$$. Numerator and denominator are perfect squares. Correct.

16 x 10$$^{-1}$$ = 1.6, or 1$$\frac{6}{10}$$, or $$\frac{16}{10}$$, whose denominator is not a perfect square. Incorrect.

16 x $$10^0$$ = 16 * 1 = 16. 16 = 4*4. Correct.

16 * $$10^1$$ = 160. Not a perfect square. Incorrect.

16 * 10$$^2$$ = 1600. 40*40 = 1600. Correct

16 * 10$$^3$$ = 16 * 1000 . . . 1000 is not a perfect square. Incorrect.

3. p = -2, 0, and 2. There are three integer values of p that will make N a perfect square.

brianne5 , I have the same question as you do, though I answered the question as if the definition of perfect square were expanded beyond integers. I wonder if GMAT adheres to the definition of perfect square as the product of a rational number multiplied by itself? Even in the Math Theory section on the forum, a perfect square is defined as "A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is an perfect square." https://gmatclub.com/forum/math-number-theory-88376.html

Yup!! Sorry.. missed that. But its -P right? So, when P=2, N=16*10^(-2)?

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:23
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Only integers can be perfect squares (e.g. see wikipedia: Square number). The answer to this question should be "two", since only p = 0 and p = -2 give us an integer which is the square of another integer. 16/100 is not a 'perfect square'.
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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:24
IanStewart wrote:
Only integers can be perfect squares (e.g. see wikipedia: Square number). The answer to this question should be "two", since only p = 0 and p = -2 give us an integer which is the square of another integer. 16/100 is not a 'perfect square'.

That's what I thought .. but it seems I was wrong !!

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:31
My solution:
1. p has to be -3, -2, -1, 0, 1, 2, or 3
2. to simplify calculations: N = 16 × 10^(-p) = 2^4 × 2^(-p) × 5^(-p) = 2^(4-p) × 5^(-p)

if p = 0 then 2^(4-0) × 5^(-0) = 2^4 = 16 = perfect square
then it's obvious even from just looking at simplification 2^(4-p) × 5^(-p) that odd values of P won't formulate perfect square.
thus, we have to check the rest 2 ( -2 and 2)

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:33
SOUMYAJIT_ wrote:
genxer123 wrote:
SOUMYAJIT_ wrote:
B.

-2 & 0.

2 will make it 4^2*10^-2.. which isn't a perfect square ? What am I missing ?

Not sure how you got bolded part. If -p = 2, we get p = -2 10$${-2}$$. 16 * 10$$^{-2}$$ = .16 = .4 * .4

Yup!! Sorry.. missed that. But its -P right? So, when P=2, N=16*10^(-2)?

Good heavens - (Neg)(Neg) was in my brain and apparently didn't make it to my typing. Nice catch!

If -p = (-2), we have -(-2) = 2. That works. 1600 = 40*40

If -p = (2), we have p = -2. That also works. .16 = .4 *.4

So I wonder if you decided that the definition of perfect square included only integers ...

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:48
SOUMYAJIT_ wrote:
That's what I thought .. but it seems I was wrong !!

You were right - the question is wrong.

That said, I don't think I've ever seen a real GMAT question that uses the phrase "perfect square". Instead, because in advanced math sometimes the definition of a perfect square is extended in various ways, the GMAT will say "square of an integer" so there is no danger of misunderstanding.
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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 08:51
IanStewart wrote:
Only integers can be perfect squares (e.g. see wikipedia: Square number). The answer to this question should be "two", since only p = 0 and p = -2 give us an integer which is the square of another integer. 16/100 is not a 'perfect square'.

hope that helps, although I do agree that some resources have the same explanation of perfect square as yours, and that was a confusion for me as well.
http://www.learnalberta.ca/content/memg ... index.html
0.64 is a perfect square
9/25 is a perfect square as well

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 09:01
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I know you'll find definitions of 'perfect square' that use an extended definition of the term, because technically, a perfect square within a given number system is just a square in that number system. A number like 3/7 is a 'perfect square' in the system of real numbers (as is any other non-negative real number), while 16/25 is a perfect square over the rational numbers, and 16 is a perfect square over the integers.

But for the purposes of GMAT math, if the phrase 'perfect square' were to appear in a real GMAT question, I can guarantee it would mean 'square of an integer'. That's the accepted definition of the phrase at the level of mathematics the GMAT is testing. This discussion is all academic though, since I don't think the GMAT would ever use the phrase, precisely because it could be unclear what definition was intended.
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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 09:11
IanStewart wrote:
SOUMYAJIT_ wrote:
That's what I thought .. but it seems I was wrong !!

You were right - the question is wrong.

That said, I don't think I've ever seen a real GMAT question that uses the phrase "perfect square". Instead, because in advanced math sometimes the definition of a perfect square is extended in various ways, the GMAT will say "square of an integer" so there is no danger of misunderstanding.

Ok!! That's a relief !

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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11 Jul 2017, 17:33
To wake up to all these replies was such a relief! Thanks for clarifying and editing the question everyone

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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23 Aug 2017, 11:02
Bunuel Please state the definition of "perfect square" as per gmat.

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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23 Aug 2017, 11:18
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rever08 wrote:
Bunuel Please state the definition of "perfect square" as per gmat.

Perfect square is an integer that is the square of an integer but as Ian Stewart mentions above GMAT most probably will use "square of an integer" instead.
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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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24 Aug 2017, 13:28
I think the fastest way to approach this problem is to realize that perfect squares have even powers in their prime factors.

So, we see that 16 gives us 4^2, now we have a 10. If we square ten, now the numbers are all even powers, so that must be a perfect square. Additionally, 10^0 = 1, which leaves us with 16, which is a perfect square. All of the other options either 1) give us odd powers, or 2) are negative and give a fraction.

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values [#permalink]

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24 Aug 2017, 23:27
Bunuel wrote:
rever08 wrote:

Perfect square is an integer that is the square of an integer but as Ian Stewart mentions above GMAT most probably will use "square of an integer" instead.

So we cannot take 16 * 10^-2 as a perfect square?

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Re: If N = 16 × 10^(-p) and −4 < p < 4, how many different integer values   [#permalink] 24 Aug 2017, 23:27

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