If m

Question

If m < p < r < t < z and if the product mprtz is positive, which of the following products must be positive?

I.  mp 
II.  rt 
III.  tz

A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III

Bunuel · Accepted Answer

Given that the product of five numbers, m , p , r , t , and z , 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 m < p < r < t < z , we can have the following three cases: m\; |\; p\; |\; r\; |\; t\; |\; z + | + | + | + | + (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. mp>0 This option is true for each of the three cases. Therefore, this option is always true. II. rt>0 This option is true for each of the three cases. Therefore, this option is always true. III. tz>0 If we have the third case, then this option is not true. Eliminate. Consequently, only options I and II are always true, and thus the answer is C. Answer: C.

tgsankar10 · Answer

for  mprtz  to be positive there must be an even number of negative numbers

The following are the possibilities for  m < p < r < t < z 
 1. 
 -m*-p*r*t*z

mp, rt  &  tz  are positive

2. 
 -m*-p*-r*-t*z

mp, rt  are positive, but  tz  is negative.

So  mp, rt  must be positive

Answer is C

DreSanson · Answer

mprtz is positive , therefore, you have 3 scenarios:

1) all 5 are positive
2) 2 lowest are negative, 3 positive.
3) 4 lowest are negative, 1 positive.

I. mp
 The 2 lowest, or are both positive (scenario 1), or both are negative (scenario 2 & 3), therefore their product will be  positive .

II. rt  
The 3rd and 4th term. They are both positive (scenario 1 & 2) or both are negative (scenario 3), therefore their product will be  positive .

III. tz
4th and 5th term. They are both positive (scenario 1 & 2), but in scenario 3, t may be negative, and z will always be positive, resulting in a scenario where their product  may be negative.

The answer will be C. I and II only must be positive.

Shakira876 · Answer

karishmaB Could you pls explain this why cant M be negative and P be positive?

DreSanson · Answer

P > M

Therefore, it isnt possible P to be negative, and M not.

It cant be M to be negative, and P to be positive, because m*r*p*z*t product is positive, it means you either have zero negatives, 2 negatives, or 4 negatives. So, if M is negative, P must also be negative.

Fiji2000 · Answer

KarishmaB  @Bunnuel

this is a ' must be true ' question right so that means we have to pick scenarios that is 100 percent the truth.

if all 5 are positive which is also a valid scenario, then C won't be the answer right?

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=84661 --- Harsha GMAT-Club-Forum-yg0cf1xk.png

ManifestDreamMBA · Answer

For the product of 5 numbers to be positive, even number of numbers will be negative, if at all there are negative numbers. Note 0s are are out as the product is positive

Also, given m < p < r < t < z, we start from the left

Either both m and p are negative or m, p, r and t all are negative

Which means mp has be to be positive and rt has to be positive.

answer C

SwethaReddyL · Answer

Hi  Bunuel , here you have given that " Given that the product of five numbers,  m ,  p ,  r ,  t , and  z , is positive,  there must be an even number of negative numbers among them" 
 
 Is this a rule or an understanding? 
 Should i take it in general as - for a product of ODD NUMBERS - there must be a even number of negative among them? 
 How about for a product of EVEN NUMBERS then?

Bunuel · Answer

Don't overcomplicate it. Negative * negative = positive, so the product of an even number of negative numbers gives positive result.

vrinda6 · Answer

Bunuel  but why are we assuming that the first two values could be negative or the last three could be positive?

Since all are in multiplication, any of the (even numbered ie 2 or 4) could be negative. And since this is a must be true question- we need to know for certain?

Bunuel · Answer

Because m < p < r < t < z, the negative numbers must come first. For example, if r is negative, then m and p must also be negative. Therefore, the only possible cases are 0, 2, or 4 negative numbers at the beginning, and we check all these cases to determine what must be positive.

egmat · Answer

Hi vrinda6,

Great question, and you're right that  without  the ordering, the negatives could sit anywhere. The key is that Bunuel's cases use the one extra fact you might be skipping past:  m < p < r < t < z . That ordering locks  which  numbers can be negative.

Here's the logic.  Negatives are always the numbers on the far  left  of an ordered list, because they're the smallest. If any variable is negative, then every variable smaller than it must also be negative.

Walk through it:
- If  p  is negative, then  m  (which is smaller) must be negative too. So you can never have "m positive, p negative" - the order forbids it.
- If  r  is negative, then  m ,  p , and  r  are all negative - you can't skip over m and p.

Therefore  the negatives can only "fill in from the left." With an  even  count of them ( 0 ,  2 , or  4 ), the only allowed pictures are:
-  0  negatives:  + + + + + 
-  2  negatives:  - - + + +  (must be m, p)
-  4  negatives:  - - - - +  (must be m, p, r, t)

You can't have, say, m and r negative but p positive - that would need p to be larger than a positive number yet smaller than a negative one, which breaks the order.

Why this settles the "must be true" part:  since m and p share the same sign in  every  case,  mp  is always positive. Same for  rt . But  tz  flips negative in the  4 -negative case, so it fails. That's C.

Quick check to lock it in:  take the ordered set  {a < b < c}  with product positive. Which spots can the  two  negatives occupy? Only  a  and  b  - never a and c. Try to place them anywhere else and the ordering breaks. Same rule, smaller set.

Answer: C

If m < p < r < t < z and if the product mprtz is positive, which of

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