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

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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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.

Answer C

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.

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

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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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)

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 ...

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.

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.
https://www.learnalberta.ca/content/memg ... index.html
https://frhaythorne.ca/eteacher_download/4221/86950
0.64 is a perfect square
9/25 is a perfect square as well

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.

Ok!! That's a relief !


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To wake up to all these replies was such a relief! Thanks for clarifying and editing the question everyone

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.

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.

Answer: B

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

A perfect square is an integer that is the square of an integer and since \(16 * 10^{-2}=\frac{16}{100}=0.16\neq integer\), then it's not a perfect square.