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If a, b, c, and d are integers and ab2c3d4 > 0, which of the

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If a, b, c, and d are integers and ab2c3d4 > 0, which of the  [#permalink]

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New post 27 Jul 2012, 09:10
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If a, b, c, and d are integers and \(ab^2c^3d^4 > 0\), which of the following must be positive?

I. \(a^2cd\)
II. \(bc^4d\)
III. \(a^3c^3d^2\)

A) I only
B) II only
C) III only
D) I and III
E) I, II, and III
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Re: If a, b, c, and d are integers and ab2c3d4 > 0, which of the  [#permalink]

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New post 27 Jul 2012, 09:29
superpus07 wrote:
If a, b, c, and d are integers and \(ab^2c^3d^4 > 0\), which of the following must be positive?

I. \(a^2cd\)
II. \(bc^4d\)
III. \(a^3c^3d^2\)

A) I only
B) II only
C) III only
D) I and III
E) I, II, and III


Since given that \(a*b^2*c^3*d^4 > 0\), then we know that none of the unknowns is zero. Therefore, \(b^2>0\) and \(d^4>0\), which means that we can safely reduce by them to get \(a*c^3>0\) (so, the given expression does not depend on the value of \(b\) or \(d\): they can be positive as well as negative).

Next, \(a*c^3>0\) means that \(a\) and \(c\) must have the same sign: they are either both positive or both negative.

Evaluate each option:

I. \(a^2cd\). Since \(d\) can positive as well as negative then this option is not necessarily positive.

II. \(bc^4d\). Since \(d\) can positive as well as negative then this option is not necessarily positive.

III. \(a^3c^3d^2\). Since \(a*c^3>0\), then \(a^3*c^3>0\) and as \(d^2>0\), then their product, \((a^3*c^3)*d^2\) must be positive too.

Answer: C.
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Re: If a, b, c, and d are integers and ab2c3d4 > 0, which of the  [#permalink]

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New post 31 Jul 2014, 21:32
a*(b^2)*(c^3)*(d^4) > 0,

since b^2 and d^4 are always positive..

a*(c^3) is positive .. so 'a' and 'c' are of the same sign, eithr positive or negative. we have no information about 'b' and 'd' , so they could b negative..

so the first 2 statements ( I and II ) have 'd' in them, which could be negative.. so we can eliminate them then and there.

in statement III, 'd' is squared, so thats positive and since 'a' and 'c' are of the same sign, a^3 * c^3 must be positive.

so III only , that option C.
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Re: If a, b, c, and d are integers and ab2c3d4 > 0, which of the  [#permalink]

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New post 08 Feb 2018, 16:40
1
superpus07 wrote:
If a, b, c, and d are integers and \(ab^2c^3d^4 > 0\), which of the following must be positive?

I. \(a^2cd\)
II. \(bc^4d\)
III. \(a^3c^3d^2\)

A) I only
B) II only
C) III only
D) I and III
E) I, II, and III


We are given that a(b^2)(c^3)(d^4) > 0.

Since b^2 and d^4 are positive, we see that the product of a and c^3 must be positive, so a and c are either both positive or both negative.

Let’s analyze each Roman numeral.

I. a^2cd

While a^2 is positive, we don’t know the sign of either c or d, so we can’t determine whether a^2cd is positive.

II. bc^4d

While c^4 is positive, we don’t know the sign of either b or d, so we can’t determine whether bc^4d is positive.

III. a^3c^3d^2

Since a and c are either both positive or both negative, a^3 and c^3 will be also either both positive or both negative. Therefore, the product a^3c^3 will be positive. Furthermore, d^2 is positive, and so a^3b^3d^2 will be positive.

Answer: C
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Re: If a, b, c, and d are integers and ab2c3d4 > 0, which of the  [#permalink]

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New post 29 Aug 2018, 17:27
I think we can realize real quick that this question is about identifying what variable might be negative and what has to be positive. We know that the given variables with their exponents will result in a positive. i.e. larger than 0. we know "a" has to be positive since we don't know if "b" might be negative or not. C is positive because the exponent is an odd exponent, and we don't know if "d" is a positive or negative. With this information, lets go down the answer choice.

Hmm, there is an issue with option 1. Because we don't know if "d" is a positive or negative, c*d could be a negative. so thats out.

Same principle with option 2. We don't know if "d" is negative or positive, so thats out too.

Which leaves us with only option 3, but lets take a quick look. We know "a" and "c" is positive, so the odd exponent is no issue, and since "d" has an even exponent, we know for sure that it must be positive.
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Re: If a, b, c, and d are integers and ab2c3d4 > 0, which of the   [#permalink] 29 Aug 2018, 17:27
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