Do You Make These 3 Mistakes in GMAT Even-Odd Questions? – Part 1/3

by on September 17th, 2015

If j is a positive integer, is (j^3-27)^2(j^3+1)^3 odd?

1)   j + 2 is even

2)   2j is even

As soon as you read the above problem, did you start thinking of the algebraic formula you can use to solve it? If yes, then you just fell in to the number #1 pitfall that stops many students from becoming an even odd champion. That’s true! Even though Even-Odd numbers is deemed to be among the easier concepts on GMAT Quant, many students are not able to solve 700+ level questions from it. Read on to find why that is the case.

In our efforts to help students ace this concept, we closely studied the mistakes that students make in Even-Odd questions.

Our research ground comprised of the doubts students ask in our internal forums, the mistakes made by 1000+ students in our recurring Number Properties Live Session, and most recently, the 5000+ attempts made on The E-GMAT Number Properties Knockout.

Our analysis shows that following are the three main pitfalls in which students tend to fall while solving Even-Odd questions:

  1. Getting intimidated by complex expressions
  2. Wasting time on unimportant terms
  3. Getting stumped in division

In this article series, we will explain each of the above 3 pitfalls with examples, discuss why it is important to avoid that pitfall, tell you how to avoid it, and finally, give you 700+ level practice questions.

Sounds good? Right then … let’s focus on Pitfall #1, which is theme of this article.

Pitfall # 1 Getting intimidated by seemingly complex expressions

What do we mean?

A few Even-Odd questions may have scary-looking expressions. For example, consider the expression given above:

If j is a positive integer, is (j^3-27)^2(j^3+1)^3 odd?

Did you too feel a bit nervous reading this question? Well, that is the first pitfall that you have to guard against. Because, if you let yourself become nervous, you will:

i) Either randomly guess the question

ii) Or panic; panic clouds our ability to think rationally, increasing our chances of making an error.

  • For example, in your panic, you may scramble to remember and apply the formula for a^3 + b^3 on the terms of this expression, only to realize that you’ve actually complicated the question.

So, as you can see, getting intimidated by complex expressions is indeed a dangerous pitfall.

What can you do to avoid this pitfall?

The next time you face such a question and notice your heartbeat increasing, take a deep breath and tell yourself,

“Since this is a GMAT question, it can be simplified elegantly.”

This is true! The beauty of official GMAT questions is that no matter how complex they look, they can always be simplified to a couple of cases.

Example: Let’s think through the question we posed above and see how it can be simplified.

1st Simplification

The given expression is (j^3-27)^2(j^3+1)^3

You’re probably familiar with the property that the power of a number doesn’t impact the even-odd nature of the number.

  • (Even)^n, where n is a positive integer = Even
  • Similarly, (Odd)^n = Odd

So,

i) (j^3- 27)^2 will have the same even-odd nature as (j^3 -27). Similarly, (j^3 + 1)^3 will have the same even-odd nature as (j^3+1)

ii) j^3 will have the same even-odd nature as j itself.

So, using this property, we’ve done the first level of simplification: now, we only have to determine the even-odd nature of this, simpler expression: (j-27)(j + 1)

2nd Simplification

The simpler expression above is a product of 2 terms: (j-27) and (j+1)

When will the product of 2 terms be odd? Only if both the 2 terms are themselves odd. If even one of these terms is even, the product will be even.

So, to answer the question, we need to know: is each of the 2 terms odd?

So, from the earlier situation of dealing with the product as a whole, we are now dealing with individual terms only: (j -27) and (j + 1)

Getting to the answer

Now, j can either be Even or Odd.

Case 1: j is odd

In this case, j + 1 = Odd + Odd = Even

And j - 27 = Odd - Odd = Even

Since both the terms are Even, the answer in this case will be NO, the given expression in not odd.

Case 2: j is even

In this case, j + 1 = Even + Odd = Odd

And, j - 27 = Even - Odd = Odd

Since both the terms are odd, the answer in this case will be YES, the given expression is odd.

So, as you can see, using this step-wise approach, we’ve been able to simplify the question to this:

Is j even?

Takeaway

  1. Don’t get intimidated by complex expressions in Even-Odd questions.
  2. Have the confidence that all Even-Odd questions in the GMAT can be easily simplified.
  3. Use the properties of Even-Odd combinations to simplify scary-looking expressions.
  4.  Avoid the impulse to search for algebraic formulae to apply on such expressions.

Test Yourself

You’ll know that you’ve learnt this lesson well if your heart doesn’t skip a beat at the first look of the following question:

If x = p*n^k+p, where n and k are positive integers, is x divisible by 2?

1. n + kn = 915
2. p^35 + 35^p is Even

Post your responses for the above question here.

Next Steps

Try the Even-Odd concept file in the Quant Free-Trial by registering for free on e-GMAT.

8 comments

  • Is it A?

    • Do we have a solution to this? I got the following:

      1) if n+kn=915 => n(1+k)=Odd => n=Odd and 1+k=Odd => 1+k => k=Even
      Therefore, n(1+k) = Odd equates to Odd(1+Even) = Odd(Odd) which equals Odd

      That tells us about n & k, but nothing about p, so NS

      2) p^35+35^p=Even, for this to be true this means Even/Odd+Even/Odd=Even, For p^35 and 35^p, you'll always get an Odd value, regardless if P is Odd or even. So we cannot conclude what P is.  At the end we have no info for p, n or K. NS.

      1 & 2 - we know what n & k are, but p =Odd or Even. When p= odd  x is divisible by 2, however when p is even X is not divisible. NS.

      Answer is E. 
      .
      This took me significantly more than 2mins to solve and I even had to use a freaking calculator for the 2nd equation, which is mad frustrating if i'm not right. 
       

    • (1): N=odd, K=even which you proved. Now, the question is whether x=(p*n^k +p) is even? Taking 'p' common from the expression gives us x=p(n^k + 1), n is odd therefore n^k is odd, TF (n^k+1) is even. This gives us x=P*(even no.) = even (whatever might P be.). Therefore x is divisible by 2. 
      which concludes (1) is sufficient.

      TF yes, ans. is A.

  • (1). n(k+1) = odd.

    Therefore, n=odd and k+1 = odd. So k =even.

    P=unknown. Thus NS.

    (2). P^35 + 35^P = Even.

    Power of a number doesn’t impact the odd/even nature of the number. So simplify the equation to P + 35 = Even. Thus, P= Odd. N and k (doesn’t matter if it’s even or odd) are unknown so NS.

    (1)+(2): n=odd, k = even, and P = odd.

    So X= P(n^k+1) = odd x (odd + 1) = EVEN = divisible by 2.

    Answer is C

    • (1): N=odd, K=even which you proved. Now, the question is whether x=(p*n^k +p) is even? Taking 'p' common from the expression gives us x=p(n^k + 1), n is odd therefore n^k is odd, TF (n^k+1) is even. This gives us x=P*(even no.) = even (whatever might P be.). Therefore x is divisible by 2. 
      which concludes (1) is sufficient.

      TF ans is A.

  • C

  • Hello Folks! :)  

    The correct answer to the exercise question is A. 

    We will post the detailed solution soon! :)

    Cheers,
    Team e-GMAT

  • This is a long but detailed explanation.  Let's assume that you are not familiar with Even & Odd rules.

    Okay, let's start out with a few Even & Odd rules. Namely,

    1)  Addition & Subtraction
     E±E=E
    O±O=E
    E±O=O

    2)  Multiplication
      E*E=E
    O*O=O
     E*O=E

    3)  Positive and negative numbers do not affect the Even & Odd results.

    4)  Positive exponents do not affect the Even/Odd of the base.  In other words, no matter the positive exponent, the result will be Even or Odd the same as the base number.

    That's all you need to know.  Now, on to the problem.

    ==========
    If x=p*n^k+p  where n and k are positive integers, is x divisible by 2?
    1.  n+kn=915
    2.  p^35+35^p is Even

    The question stem is asking, is x even or does p*n^k+p give an even result (any number divisible by 2 is an even number).  So, the only set up options to get a YES (an even result) must look like this.

    (p*n^k) + p = ?
        (E*E) + E = E
       (O*O) + O = E
        (E*O) + E = E

    And, the only set up option to get a NO (an odd result) will look like this.
    (p*n^k) + p = ?
       (O*E) + O = O

    [NOTE: According to rule #4 k can be ignored.]

    In other words, if p is even then n can be even or odd.  But if p is odd then n must be odd too, in order to get a YES (an even result).

    Now, for the statements.

    ==========
    Statement 1)   n+kn=915

       Translated:  n+kn = odd
     Use rule #1:  E+O=O  or  O+E=O

    Therefore, n+(kn)=915 must look like this to be true:
       O+(E*O) = O
    Which means that n=odd and k=odd

    Plug this back into the question stem problem and we get
       p*(n^k)+p = ?
      p*(O^O)+p = ?
    Now remember rule #4 from above.  The problem will now look like
      p*O+p = ?

    The answer will always be Even because it does not matter whether p is odd or even.  Consider this,
    (O*O)+O=E  =  (O)+O=E
     (E*O)+E=E  =  (E)+O=E

    [NOTE: According to rule #3 it does matter whether p is positive or negative]

    Statement 1 is SUFFICIENT
      Answer: A or D

    Alternatively,

    ==========
    Statement 1)   n+kn=915

       Converted:  n(1+k)=915
       Translated:  n(1+k) = odd
     Use rule #2:  O*O=O
        Therefore:  n=odd -- (1+k)=odd -- k=even

    Plug this back into the question stem problem and we get
      p*(n^k)+p
      p*(O^E)+p = ?
    Now remember rule #4 from above.  The problem will now look like
      p*O+p = ?

    The answer will always be Even because it does not matter whether p is odd or even.  Consider this,
    (O*O)+O=E  =  (O)+O=E
     (E*O)+E=E  =  (E)+O=E

    [NOTE: According to rule #3 it does matter whether p is positive or negative]

    Statement 1 is SUFFICIENT
      Answer: A or D

    ==========
    Statement 2)   p^35+35^p is Even

    Firstly, remember rule #4 above and we know that p will be even or odd regardless of its exponent and 35 will remain odd regardless of the value of p.

    So statement 2 translates from
      p^35+35^p = Even
        to
      p+O=E

    Remember rule #1 above and we know that the only way this is true is if p is odd:
      O+O=E

    Plug this back into the question stem problem and we get
      p*(n^k)+p
      O*(n^k)+O = ?

    The only set up for p*(n^k)+p to be even is n is odd (remember rule #4 above and we can ignore k).
      O*(n^k)+O = ?
         O*(O)+O = E

    But if n is even then the result will be odd.
      O*(n^k)+O = ?
          O*(E)+O = O

    And since we do not have any information about n and n could be even or odd, then:

    Statement 2 is NOT SUFFICIENT
      Answer: D

    It took me several, several practice problems to finally memorize the Even & Odd formulas, but I've got it now and I look at these problems as automatic correct answers.  Thanks for the freebies, GMAT.

    Problems like this may seem complex but once you get the hang of Even & Odd rules and you can visualize it all in your head then it's easy.

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