Numbers with powers

harsh.champ wrote:If U, V, W and m are natural numbers such that (U^m) +( V^m) = (W^m), then which of the following is true?


(1)m < Min (U, V, W)
(2)m > Max (U, V, W)
(3)m < Max (U, V, W)
(4)None of the above

What's the source? There are no GMAT questions with only 4 choices, so you're obviously getting this from somewhere else. The math seems to be way above the level of the GMAT.

Ignoring that for a moment, we can quickly narrow it down to 2 choices.

First, even if we don't understand what's going on, we should be able to eliminate (1), since if (1) is true, (3) will also always be true. Both (1) and (3) can't be the right answer, so eliminate (1).

Second, we can see that if m=2, we simply have the Pythagorean Theorem, a^2 + b^2 = c^2. The smallest natural numbers that fit are 3, 4, 5. Clearly m (2) isn't greater than the Max (5), so eliminate (2) as a "must be true".

Trying to determine whether (3) is always true is WAY beyond the scope of the GMAT, so feel free to stop reading here, but let's give it a shot.

The first major issue is the definition of the set of Natural numbers. I only have a copy of OG10 at home, but I don't think any of the math definitions have changed. The set of Natural numbers isn't defined in the OG or in any GMAT materials that I've ever seen; in fact, I don't think I've see that term ever actually used on the GMAT, for good reason: there are two accepted definitions.

There's only one difference between the two, but it's a key difference to this question: whether 0 is a member of the set.

In the first definition, the set includes only positive integers; in the second definition, it includes all non-negative integers.

If we follow the first, i.e. 0 is not included, then (3) is the correct answer to the question, since making m greater than or equal to W (the biggest number out of U, V, W) will be impossible.

However, if we accept the second definition, i.e. 0 is included, then we can let U, V and W all equal 0 and pick any non-0 m we choose. For example:

0^100 + 0^100 = 0^100

is true.

(Nowhere does it say that U, V and W must be distinct.)

Accordingly, if we include 0 in the set of Natural numbers, the answer would be (4).

So, not only is this question un-GMATesque, it's also unanswerable without more information.

Stuart Kovinsky wrote:

harsh.champ wrote:If U, V, W and m are natural numbers such that (U^m) +( V^m) = (W^m), then which of the following is true?


(1)m < Min (U, V, W)
(2)m > Max (U, V, W)
(3)m < Max (U, V, W)
(4)None of the above

What's the source? There are no GMAT questions with only 4 choices, so you're obviously getting this from somewhere else. The math seems to be way above the level of the GMAT.

Ignoring that for a moment, we can quickly narrow it down to 2 choices.

First, even if we don't understand what's going on, we should be able to eliminate (1), since if (1) is true, (3) will also always be true. Both (1) and (3) can't be the right answer, so eliminate (1).

Second, we can see that if m=2, we simply have the Pythagorean Theorem, a^2 + b^2 = c^2. The smallest natural numbers that fit are 3, 4, 5. Clearly m (2) isn't greater than the Max (5), so eliminate (2) as a "must be true".

Trying to determine whether (3) is always true is WAY beyond the scope of the GMAT, so feel free to stop reading here, but let's give it a shot.

The first major issue is the definition of the set of Natural numbers. I only have a copy of OG10 at home, but I don't think any of the math definitions have changed. The set of Natural numbers isn't defined in the OG or in any GMAT materials that I've ever seen; in fact, I don't think I've see that term ever actually used on the GMAT, for good reason: there are two accepted definitions.

There's only one difference between the two, but it's a key difference to this question: whether 0 is a member of the set.

In the first definition, the set includes only positive integers; in the second definition, it includes all non-negative integers.

If we follow the first, i.e. 0 is not included, then (3) is the correct answer to the question, since making m greater than or equal to W (the biggest number out of U, V, W) will be impossible.

However, if we accept the second definition, i.e. 0 is included, then we can let U, V and W all equal 0 and pick any non-0 m we choose. For example:

0^100 + 0^100 = 0^100

is true.

(Nowhere does it say that U, V and W must be distinct.)

Accordingly, if we include 0 in the set of Natural numbers, the answer would be (4).

So, not only is this question un-GMATesque, it's also unanswerable without more information.

____________________________
I got this question from a tutor at my native place.
You wrote that these type of questions cannot feature in the GMAT.Is it not worthwhile to practice some questions of higher level??
I also found the question a bit difficult but that tutor said that practicing tough questions helps you open up your mind.
Seeking advice!!

harsh.champ wrote:If U, V, W and m are natural numbers such that (U^m) +( V^m) = (W^m), then which of the following is true?


(1)m < Min (U, V, W)
(2)m > Max (U, V, W)
(3)m < Max (U, V, W)
(4)None of the above

U=0 V=1 and W=1 and m=100

1) doesn't hold
3) Doesnt hold

U=3; V=4 and W=5 with m =2

2) doesnt hold

so I would say 4