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Bunuel
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If a > b > c > d > e and abcde > 0, then which of the following must 27 Jan 2022, 02:12
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55% (hard)

Question Stats:

65% (01:40) correct 35% (01:41) wrong based on 409 sessions

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If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)

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

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Bunuel
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If a > b > c > d > e and abcde > 0, then which of the following must 06 Apr 2023, 03:55
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MissionAdmit
Hi Bunuel can you please help with the answer? None of the options is a must true, I could figure out ways in which all options fail

I
ab>0 , either both a and b positive or both negative. Assume e is negative while c and d is positive then whole abcde<0

II
bc>0
Assume both b and c are negative. Now a can be negative and still a>b>c, also d and e will definitely be negative. Therefore abcde<0

III
de>0
Assume both negative, c can be negative, and still c>d>e. Also, a and b can be positive, and a>b>c. Therefore abcde<0

Bunuel
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)

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


FRESH GMAT CLUB TESTS' QUESTION

Are You Up For the Challenge: 700 Level Questions
Show more

Official Solution:


If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)

II. \(bc > 0\)

III. \(de > 0\)


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


Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:

\(a\; |\; b\; |\; c\; |\; d\; |\; e\)

\(+ | + | + | + | +\) (no negative numbers)

\(+ | + | + | - | -\) (two negative numbers)

\(+ | - | - | - | -\) (four negative numbers)

Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.

I. \(ab > 0\)

If we have the third case, then this option is not true. Eliminate.

II. \(bc > 0\)

This option is true for each of the three cases. Therefore, this option is always true.

III. \(de > 0\)

This option is true for each of the three cases. Therefore, this option is always true.

Consequently, only options II and III are always true, and thus the answer is D.


Answer: D
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NKJJ1990
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If a > b > c > d > e and abcde > 0, then which of the following must 27 Jan 2022, 03:37
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a > b > c > d > e
a b c d e > 0
=> There have to be an even number of negatives, i. e., either 2 or 4 negative signs.
a | b | c | d | e
+ | + | + | + | +
+ | + | + | - | -
+ | - | - | - | -

ab > 0; No
It may or may not be. +*+ = +; +*- = -

bc > 0; Yes

de > 0; Yes

Answer II and III
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If a > b > c > d > e and abcde > 0, then which of the following must 05 Apr 2023, 13:51
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Hi Bunuel can you please help with the answer? None of the options is a must true, I could figure out ways in which all options fail

I
ab>0 , either both a and b positive or both negative. Assume e is negative while c and d is positive then whole abcde<0

II
bc>0
Assume both b and c are negative. Now a can be negative and still a>b>c, also d and e will definitely be negative. Therefore abcde<0

III
de>0
Assume both negative, c can be negative, and still c>d>e. Also, a and b can be positive, and a>b>c. Therefore abcde<0

Bunuel
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)

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


FRESH GMAT CLUB TESTS' QUESTION

Are You Up For the Challenge: 700 Level Questions
Show more
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dhwanit2412
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If a > b > c > d > e and abcde > 0, then which of the following must 11 Jun 2024, 10:31
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But why can't we assume that all the numbers are positive so everything must be correct. this is a question filled with errors. Please remove this question down.
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sourabh23
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If a > b > c > d > e and abcde > 0, then which of the following must 11 Jun 2024, 10:44
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dhwanit2412
But why can't we assume that all the numbers are positive so everything must be correct. this is a question filled with errors. Please remove this question down.
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the question is correct. it asks you which of the following “must be true” i.e there are variables in the question but the answer must confirm to all the possibilities of the given variables (as is explained in the answer)

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If a > b > c > d > e and abcde > 0, then which of the following must 02 Mar 2025, 04:35
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What if we take b c d e as a decimal number? Official answer logic wont be valid here.
Bunuel
MissionAdmit
Hi Bunuel can you please help with the answer? None of the options is a must true, I could figure out ways in which all options fail

I
ab>0 , either both a and b positive or both negative. Assume e is negative while c and d is positive then whole abcde<0

II
bc>0
Assume both b and c are negative. Now a can be negative and still a>b>c, also d and e will definitely be negative. Therefore abcde<0

III
de>0
Assume both negative, c can be negative, and still c>d>e. Also, a and b can be positive, and a>b>c. Therefore abcde<0

Bunuel
If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)
II. \(bc > 0\)
III. \(de > 0\)

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


FRESH GMAT CLUB TESTS' QUESTION

Are You Up For the Challenge: 700 Level Questions

Official Solution:


If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)

II. \(bc > 0\)

III. \(de > 0\)


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


Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:

\(a\; |\; b\; |\; c\; |\; d\; |\; e\)

\(+ | + | + | + | +\) (no negative numbers)

\(+ | + | + | - | -\) (two negative numbers)

\(+ | - | - | - | -\) (four negative numbers)

Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.

I. \(ab > 0\)

If we have the third case, then this option is not true. Eliminate.

II. \(bc > 0\)

This option is true for each of the three cases. Therefore, this option is always true.

III. \(de > 0\)

This option is true for each of the three cases. Therefore, this option is always true.

Consequently, only options II and III are always true, and thus the answer is D.


Answer: D
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If a > b > c > d > e and abcde > 0, then which of the following must 02 Mar 2025, 04:59
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KumarRishav
What if we take b c d e as a decimal number? Official answer logic wont be valid here.
Bunuel
Official Solution:


If \(a > b > c > d > e\) and \(abcde > 0\), then which of the following must be true ?

I. \(ab > 0\)

II. \(bc > 0\)

III. \(de > 0\)


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


Given that the product of five numbers, \(a\), \(b\), \(c\), \(d\), and \(e\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(a > b > c > d > e\), we can have the following three cases:

\(a\; |\; b\; |\; c\; |\; d\; |\; e\)

\(+ | + | + | + | +\) (no negative numbers)

\(+ | + | + | - | -\) (two negative numbers)

\(+ | - | - | - | -\) (four negative numbers)

Let's analyze each option, taking into consideration that the question asks which of them MUST be true, not COULD be true.

I. \(ab > 0\)

If we have the third case, then this option is not true. Eliminate.

II. \(bc > 0\)

This option is true for each of the three cases. Therefore, this option is always true.

III. \(de > 0\)

This option is true for each of the three cases. Therefore, this option is always true.

Consequently, only options II and III are always true, and thus the answer is D.


Answer: D
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The numbers will still fall under one of the three cases mentioned in the OA, irrespective of whether they are decimal or integer, the logic in OA will remain valid.
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