chetan2u wrote:

genxer123 wrote:

Bunuel wrote:

If n^3 is odd, which of the following statements are true?

I. n is odd.

II. n^2 is odd.

III. n^2 is even.

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and III only

\(n^3\) is odd. From that we can deduce that \(n\) must be odd.

There is only one way a number, \(n\), multiplied by itself 3 times or 100 times, can be odd: \(n\) is odd.

Odd * odd is odd. Odd * even is even.

In fact, the only way to get an odd product is to have only odd factors. If there just one even term (or two or 50), the product is even.

There is an odd product? Then there are ONLY odd factors.

Which statements are true?

I: n is odd. See above. For \(n^3\) to be odd, n must be odd. TRUE

II. \(n^2\) is odd. Odd * odd = ODD. TRUE

III. \(n^2\) is even. If n is odd, as it is here, \(n^2\) can never be even. Odd * odd = odd. FALSE

Answer D

hi..

what if n= \(\sqrt[3]{x}\) then neither n nor n^2 is ODD..

Had the Q been Could be true... YES

Although this is what it means here

chetan2u , as is often the case, you are provocative, which I appreciate and respect.

Perhaps I do not understand what the x is in your n= \(\sqrt[3]{x}\)

If x = 27, then \(\sqrt[3]{27}\) = 3

If x = 8, then \(\sqrt[3]{8}\) = 2

What cube root of an even number is odd?

More important, what even number, cubed, is odd?

I am confused.

And we

are told that n^3 is ODD.

From that fact we are asked to infer number properties.

I considered n= \(\sqrt[3]{8}\) (similar to what you are suggesting, where x = 8, and n = 2).

I decided that such an approach was to reason the wrong way:

That is, to reason as such is to reason

from a possibility for n

to the

given \(n^3\), rather than

from the

given \(n^3\) (= ODD)

to the possible properties of \(n\).

Neither does the question's imperative -- TRUE (not "could be possible" or "could be true") -- seem to allow for such reasoning.

So I decided to answer the question the way I thought it was intended, a perceived intention upon which you agree.

Also, not wanting to sound too full of myself, or too esoteric, I decided not to mention the alternative I had considered, which seemed . . . inapposite. (I'm not an expert. You are.)

It is an interesting point. Is it apposite?

Were I to see a formulation such as that which you postulate, of course my answer would be different. Thanks, and kudos for noting subtleties (even though I am slightly confused about whether or not your suggestion applies to this question and its answer).

(P.S. This group can be a tough crowd. I still would not include your possibility in my answer.)

_________________

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that within me there lay an invincible summer.

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