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Number theory

by STEVEN SPIELBERG » Sun Dec 08, 2013 8:01 am
Q:If P be a prime number such that 3<P<50, then P^2-1 is:

(a)Odd

(b)A perfect square

(c)A fraction

(d)A negative integer

(e)Always divisible by 8

__________________________________________________________________________________________________

I have few questions regarding the above question:

1) Is this a number pattern question, since we have a pattern rule P^2-1 which will always give the same result whatever value we may plug in for P from the domain ? Please don't mention number properties question which we already know it is. This question is based on elementary number theory which is mostly based on number patterns.

2)Is p^2-1 a Non-recursive(Explicit) rule/formula/definition for the number pattern since this rule satisfies a specific set of values or a number pattern ?

3)Is 3<P<50 a Domain Restriction since we are allowed to input only a specific range of numbers and that too being Prime ?

Now there are many Math Experts and many maths and science and engineering students out there who can very easily answer these questions. I request them to please answer these specific questions as I was having difficulty handling these kind of questions. Atleast I should know the correct name/nomenclature of this specific question type!

Please don't name Number properties or number theory which we know it is!
Last edited by STEVEN SPIELBERG on Sun Dec 08, 2013 9:12 am, edited 1 time in total.
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by Brent@GMATPrepNow » Sun Dec 08, 2013 8:43 am
STEVEN SPIELBERG wrote:Q:If P be a prime number such that 3 < P < 50, then P² - 1 is:

(a)Odd
(b)A perfect square
(c)A fraction
(d)A negative integer
(e)Always divisible by 8
I'd like to answer your 3 questions, but I'm not exactly sure what you're asking for each of them. Also, I'm not sure how to handle your request to refrain from "naming Number properties or number theory which we know it is"
If you want us to avoid mentioning number properties that everyone knows, you might first tell us which number properties those are, since I doubt there are any properties that everyone knows.

So, at the risk of using commonly-acknowledged properties, I'll tell you how I'd solve this.

First recognize that P² - 1 is a difference of squares that we can factor to get (P - 1)(P + 1)
Now notice that (P - 1), P and (P + 1) are consecutive integers.
Since P is a prime number that's greater than 3, we know that P IS ODD
If P is odd, then (P - 1) and (P + 1) are both EVEN
In fact, (P - 1) and (P + 1) are CONSECUTIVE EVEN integers.
Integer property: In a set of CONSECUTIVE EVEN integers, every second value will be divisible by 4.
For example, in the set {32, 34, 36, 38, 40, 42, 44, 46, 48} we can see that 32, 36, 40, 44 and 48 are all divisible by 4.
Since (P - 1) and (P + 1) are two CONSECUTIVE EVEN integers, each is divisible by 2.
More importantly, one of them must be divisible by 4.
So, either (P - 1) or (P + 1) is divisible by 2 and the other is divisible by 4. This means that their product, (P - 1)(P + 1), MUST BE DIVISIBLE by 8 (the product of 2 and 4)
In other words, P² - 1 MUST BE DIVISIBLE by 8

So, the correct answer is E

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Sun Dec 08, 2013 9:39 am, edited 1 time in total.
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by Brent@GMATPrepNow » Sun Dec 08, 2013 8:51 am
STEVEN SPIELBERG wrote:Q:If P be a prime number such that 3<P<50, then P^2-1 is:

(a)Odd

(b)A perfect square

(c)A fraction

(d)A negative integer

(e)Always divisible by 8
The best (fastest) approach here is to TEST values.

If P = 5, then P² - 1 = 5² - 1 = 24
24 is NOT odd. So, eliminate A
24 is NOT a perfect square. So, eliminate B
24 is NOT a fraction. So, eliminate C
24 is NOT a negative integer. So, eliminate D
24 IS divisible by 8.

We're left with E, so it must be the correct answer.

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Sun Dec 08, 2013 9:41 am, edited 1 time in total.
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by STEVEN SPIELBERG » Sun Dec 08, 2013 9:27 am
ooops! My mistake! the first option is actually (a)Odd. So I have corrected the error in the first post.
OA(e)

What I was asking was a very specific name for this type of question cause Number properties is a very regular name for this. Normally, we name a sequence as a number pattern .Or when a specific set of numbers satisfies an equation/rule we call it a number pattern, so similarly are these type of questions:P^2-1 etc which satisfies specific values/numbers called number patterns question ?

And rest of two questions mentioned in the first post remains same
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by Brent@GMATPrepNow » Sun Dec 08, 2013 9:35 am
STEVEN SPIELBERG wrote:ooops! My mistake! the first option is actually (a)Odd. So I have corrected the error in the first post.
OA(e)

What I was asking was a very specific name for this type of question cause Number properties is a very regular name for this. Normally, we name a sequence as a number pattern .Or when a specific set of numbers satisfies an equation/rule we call it a number pattern, so similarly are these type of questions:P^2-1 called number patterns.

And rest of two questions mentioned in the first post
It's hard to categorize questions that involve the expression P² - 1
Instead, I'd say that if a question involves the expression P² - 1, then one should definitely factor it to get (P-1) and (P+1) and then use the fact that P - 1), P and (P + 1) are consecutive integers (as long as P itself is an integer)

If I had to categorize questions that involve the expression P² - 1, then I'd say they're likely a hybrid involving divisibility rules AND consecutive integers.

Cheers,
Brent
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by Mathsbuddy » Sun Dec 08, 2013 6:57 pm
Hi there,

Brent has answered the main question very effectively, so I won't elaborate on that.
In response to your questions, I don't believe it is a "number pattern" question per se. It's just a test of logic as far as I can see. p^2-1 could be used to formulate a number pattern if you want to, but I don't see that it's of importance here. 3<P<50 is just there to eliminate "2" from the prime numbers, while restricting the top end to stop people wasting time time while infinitely testing for multiples of 8! The parameters given are there to help you choose your path; nomenclature is not required.

Eg: as 2 is not included, the only prime numbers that remain are odd. Odd * odd = odd, so subtracting one will make it even everytime.

I don't know if this helps or not, but let me know if not.
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by [email protected] » Sun Dec 08, 2013 9:56 pm
Hi STEVEN,

GMAT questions are usually written in a manner that allows for more than one approach to get to the correct answer. This design element actually rewards Test Takers who are flexible in their thinking, since certain approaches are faster than others for certain questions. Knowing more than one approach can also help a Test Taker to get "unstuck" on a trickier question.

Your goal on any given question is two-fold:
1) Get the question correct, within a reasonable amount of time
2) Use the approach that is fastest

This question is PERFECT for TESTing Values (one of Brent's explanations showed this approach). While you can certainly learn other, more complex ideas and approaches to questions, it doesn't tend to serve much purpose since you're not scored on HOW you approach the question. In that same way, taking the longer, more technical (or more step-heavy) approach will likely eat up chunks of your clock, which could cause a pacing problem. In most cases, you don't have to come up with a complex solution to answer the question, so you should factor that idea into your continuing studies.

GMAT assassins aren't born, they're made,
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by Mathsbuddy » Sun Dec 08, 2013 11:58 pm
Brent@GMATPrepNow wrote:
STEVEN SPIELBERG wrote:Q:If P be a prime number such that 3<P<50, then P^2-1 is:

(a)Odd

(b)A perfect square

(c)A fraction

(d)A negative integer

(e)Always divisible by 8
The best (fastest) approach here is to TEST values.

If P = 5, then P² - 1 = 5² - 1 = 24
24 is NOT odd. So, eliminate A
24 is NOT a perfect square. So, eliminate B
24 is NOT a fraction. So, eliminate C
24 is NOT a negative integer. So, eliminate D
24 IS divisible by 8.

We're left with E, so it must be the correct answer.

Cheers,
Brent
This might seem trivial, but I would want to test the lowest possible value before concurring that P² - 1 is NOT (always) a negative integer: 3^2 - 1 = 8. Any thoughts on this?
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by STEVEN SPIELBERG » Mon Dec 09, 2013 6:09 am
Hey guys,

yep! I agree with Rich this does not add much to my prep. Knowing theory is not that much necessary than the practical aspect of actually solving the problem does. I was just curious cause this is one type of problems that's always bothering me from my undergrad days.

Now mathsbuddy wrote that this is not a number pattern question. But when we check the values that satisfy the equation P^2-1 we get a sequence of multiples of 8. So ain't that a number pattern and ain't P^2-1 a number pattern rule/equation/definition. Also a rule can be recursive in which case the first term must be given e.g a=1 etc. and we relate back each term to a prior term. Whereas in case of a non-recursive rule no prior term is given and we can straight input the nth value to find the nth term. So ain't P^2-1 a non recursive/explicit rule/equation for the number pattern ? I was curious to know cause I have seen many questions of this type in management entrance tests and this type gets the students unsettled. Atleast knowing the correct name of the question type takes the fear factor out of it. I have many sequence based questions which are actually called number pattern question. Since this question type replicate sequence based questions ain't these be called number pattern questions?
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by Aman verma » Tue Dec 10, 2013 8:52 am
STEVEN SPIELBERG wrote:Hey guys,

yep! I agree with Rich this does not add much to my prep. Knowing theory is not that much necessary than the practical aspect of actually solving the problem does. I was just curious cause this is one type of problems that's always bothering me from my undergrad days.

Now mathsbuddy wrote that this is not a number pattern question. But when we check the values that satisfy the equation P^2-1 we get a sequence of multiples of 8. So ain't that a number pattern and ain't P^2-1 a number pattern rule/equation/definition. Also a rule can be recursive in which case the first term must be given e.g a=1 etc. and we relate back each term to a prior term. Whereas in case of a non-recursive rule no prior term is given and we can straight input the nth value to find the nth term. So ain't P^2-1 a non recursive/explicit rule/equation for the number pattern ? I was curious to know cause I have seen many questions of this type in management entrance tests and this type gets the students unsettled. Atleast knowing the correct name of the question type takes the fear factor out of it. I have many sequence based questions which are actually called number pattern question. Since this question type replicate sequence based questions ain't these be called number pattern questions?
Hello there,

This is not a number pattern question but rather a Number theory question and if you want to be more specific this is a Modular Arithmetic question. Most of the questions you see of this type is basically a Modular Arithmetic Question based on number theory but remember the limitations of modular arithmetic, you can use only integers, that means odds,evens and primes.

Hope this helps and I would like to see what other members has to say about this!
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by Abhishek009 » Tue Dec 10, 2013 9:44 am
STEVEN SPIELBERG wrote:Q:If P be a prime number such that 3<P<50, then P^2-1 is:

(a)Odd

(b)A perfect square

(c)A fraction

(d)A negative integer

(e)Always divisible by 8

P be a prime number such that 3<P<50

P = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. }

For testing purpose we don't need to find the entire set just 3 numbers can solve our purpose...


If P = 3

P^2-1 = > 8

If P = 5

P^2-1 => 24

If P = 7

P^2-1 => 40


In all the cases the result obtained is divisible by 8 ....


Hence option (e) seems to be the best among the rest...
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by Matt@VeritasPrep » Thu Dec 12, 2013 5:42 pm
Here's what's happening, as Brent said. This is all that matters - the name you choose to give the question type and how else you can solve it (i.e. some more complicated way) are irrelevant.

Primes greater than 2 are always odd, obviously.

So if p represents a prime greater than 2, p is odd.

p² - 1 = (p + 1) * (p - 1)

Since p is odd, (p + 1) is even and (p - 1) is even.

So p² - 1 = (even) * (even) = a multiple of 4.

Since (p + 1) and (p - 1) are CONSECUTIVE EVEN INTEGERS, one of them (we don't know which) is a multiple of 2 and the other is a multiple of 4.

So p² - 1 = (multiple of 2) * (multiple of 4) = (multiple of 8).

You generally shouldn't do anything recursive (or non-recursive) at the GMAT level with primes because the primes do not constitute a particularly well-behaved "sequence", and on this Q the only thing that matters about them is that they're odd.

The formal definition of a sequence can be found here, if you're curious. This requires some formal mathematical training to parse (though it sounds like you have that).
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by Aman verma » Fri Dec 13, 2013 6:33 am
Hello there,

As Matt has mentioned you don't have to bother to much about the terminology for the GMAT, but for your curiosity let me clarify few things. A Pattern rule can be recursive or non- recursive. A recursive definition is one in which the first element or any prior element is given and we find the later element using the rule and relating back to the prior element. Whereas a non-recursive definition is one in which no prior element is given, you just plug the nth term to find out the value of the nth term. This type of rule is also known as Explicit Rule.

Also,there is a sequence in this question but you don't have to go so far as pattern rule or something. Though there is generally a pattern for the prime numbers:- 6n+5,6n-1,2 or 3.

Also,most questions of this type are based on Modular Arithmetic based on Elementary Number Theory. Although, you never have to use Modular Arithmetic to solve this type of questions. You generally use number picking, backsolving or number testing to solve this type of questions. The root or source of this type of questions or most number properties questions are based on Modular Arithmetic. Most questions involving Integers such prime numbers,odds,evens,Integral factors etc are based on Modular Arithmetic.These properties can be evaluated using Caylay Table. Though you never have to use Modular Arithmetic to solve this type of questions on the GMAT. You just use number picking or backsolving to solve this type of questions.

I haven't done much math after my 12th grade. Most of the above is based on Net research. I don't have any exposure to number theory or Abstract algebra. But having said that I haven't done much math after 12th Grade what I meant that I didn't have math as a course in my undergrads. But as a Chartered Accountant student I have exposure to calculus,theoretical distributions,financial engineering and advanced maths application to finance and economics.I have some exposure to financial modelling , simulation, reverse engineering,Pert-CPM,Linear Programming,Operations Research to mention a few.
(Although all this is irrelevant for the GMAT).
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by madhavanjc » Wed Dec 18, 2013 7:10 am
Q:If P be a prime number such that 3 < P < 50, then P² - 1 is:

(a)Odd
(b)A perfect square
(c)A fraction
(d)A negative integer
(e)Always divisible by 8

I try to filter the options based on the conditions.
As P is prime, P is odd and ((p^2)-1) will be even.
eg. if p is 7 p^2=49 and p^2-1 will be 48(even).

so we can filter a and b.
The value cannot be a fraction. So filter c.
The square value cannot be negative. So filter d.
Finally the rest out answer will be e.
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