If x is an even integer, which of the following must be an odd integer

One can spot right away that if \(x\) is any even number then \(x^2\) is a multiple of 4, which makes \(\frac{x^2}{2}\) an even number and therefore \(\frac{3x^2}{2}+1=3*even+1=even+1=odd\).

Answer: E.

If you don't notice this, then one also do in another way. Let \(x=2k\), for some integer k, then:

A. \(\frac{3x}{2}=\frac{3*2k}{2}=3k\) --> if \(k=odd\) then \(3k=odd\) but if \(k=even\) then \(3k=even\). Discard;

B. \(\frac{3x}{2}+1=\frac{3*2k}{2}+1=3k+1\) --> if \(k=odd\) then \(3k+1=odd+1=even\) but if \(k=even\) then \(3k+1=even+1=odd\). Discard;

C. \(3x^2\) --> easiest one as \(x=even\) then \(3x^2=even\), so this option is never odd. Discard;

D. \(\frac{3x^2}{2}=\frac{3*4k^2}{2}=6k^2=even\), so this option is never odd. Discard;

E. \(\frac{3x^2}{2}+1=\frac{3*4k^2}{2}=6k^2+1=even+1=odd\), thus this option is always odd.

Answer: E.

Similar question to practice: even-and-odd-gmatprep-88108.html

Hope it helps.
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Well,
E+O = O
O+O = E
etc.

For this question, answer should be E.

3x^2/2 + 1

if x is even, x^2 is always even. If x is an even integer then x^2/2 is always even. E*O = E, which means 3*x^2/2 is always even. E+O = O which means 3x^2/2 + 1 is odd.

I think that with those 3 rules you have your solution already

x is E (even) and therefore x^2 is E.

3x^2 is E, as it is O*E

Therefore, 3x^2/2 will be E. (since x is integer, x^2 > 2)

If you add 1 to an E number, you will always get odd.

Hence, E is the answer.

Ah ha! That's the ticket - I forgot that if x = Even, then x^2 = Even.
I didn't quite understand the statement:

since x is integer, x^2 > 2. Did you mean x^2 > x?

I agree that the answer is E, but I see why B is confusing. I concluded in E simply b/c of process of elimination. I can disprove (b) by plugging in x=2 ==>4==>even.

I guess that's the right way to approach it.

Ok, then, is it safe to suppose then, that every EVEN number raised to a power, divided by that same base (2), must be EVEN:

2^2 = 4/2 = 2 (E)
2^3 = 8/2 = 4 (E)
2^4 = 16/2 = 8 (E)


This is opposed to an EVEN number divided by an EVEN number, which can either result in an EVEN or ODD number:

2/2 = 1
4/2 = 2
6/2 = 3
8/2 = 4

This is not true.

x = 6 for instance.



The easiest approach to this answer is to count the "minimum" even prime factors (aka 2s).

If we know X is even, we have at least one 2 as a factor of X.

If we have X^2, we double all those factors.

Thus, X^2/2 is guaranteed to be even.

Even + 1 = Odd

The reason why (3X^2/2) + 1 is even is consider just X^2/2 part.

1) First X^2/2 can be written as X.X/2
2) X is even.
3) X/2 can be Even or Odd
3) That means X.X/2 is Even*Even OR Odd number
4) This number is always Even.
5) 3*Even number is Even.
6) Even number + 1 is ODD.

There we arrive at the answer. Simple as that.

One way to choose between B and C:
n is even ---> n = 2k with k is an integer

From B: 3x/2 + 1 = 3k + 1 --> the result can be either odd or even, depending on k

From E: 3x^2/2 + 1 = 6k^2 + 1 ---> always odd
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+1 for the precise explanation

In option B, given that x is 6 (2 x 3=6) for example, the 2 can be cancelled out so that 3x/2 results in 9, which is an odd integer. 9+1=10, which is even. However, once there are more than one 2s, the fraction will always result in an even number and finally add up to an odd number.

Since x is even, x^2 must be even also. Furthermore, x^2 must be a multiple of 4 since it will have at least two factors of 2. Therefore, 3x^2/2 is even and since even + 1 = odd, 3x^2/2 + 1 will always be odd.

Answer: E
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Such a question on GMAT tests your skills on Number Properties and that too specifically from the topic of Evens and Odds.
I would always suggest you to learn using SMART plugin strategies but alongside also know the concepts that can help you sail through easily.
GMAT Concept on Evens & Odds.

GMAT Concept Tested here

Even numbers are those that can be divided evenly by 2, while odd numbers are those that cannot. Examples include {……,-10, -8, -6, -4, -2, 0 , 2, 4, 6 ,8, 10 …….}
For example, 4 is even because it can be divided by 2 without a remainder, but 5 is odd because 3 is not a whole number and therefore cannot be used to divide 5 evenly. 6, 8, 10, 12.
Even Numbers thus can be expressed as 2n where n is an integer and odd numbers can be expressed as 2n + 1 or 2n – 1.

Operations on Evens & Odds-
Many questions on GMAT can be solved using operations on Evens and Odds.

The following are the properties that can be used

1. An even number added or subtracted with an even and an odd number added or subtracted with an odd number will ALWAYS generate an even number whereas an even added or subtracted with an odd number will always generate an odd number.

2. Similarly, an even number multiplied with an odd or even will always generate an even number whereas an odd number multiplied with an odd number will always generate an odd number.

3. An even number divided by an even number doesn’t give us any definite conclusion. For example, 6 / 2 = 3 which is an odd number but 4/2 =2 which is an even number. An odd number divided by another odd number will give us an odd value as the result if the result is an integer.


Summarizing-
Addition OR Subtraction
even +/- even = even
even +/- odd = odd;
odd +/- odd = even.

Multiplication
even * even = even
even * odd = even

Now getting back to the question

How would a SMART GMAT aspirant think through here?

GMAT Track of thought 1

The answer choices are all in the form of options carrying variables. Can I use some simple values and test the choices? Let me pick up x = 0 since 0 is the simplest even I can think of for the problem. Since it’s a “Must be True” type of GMAT question, the option must be ALWAYS true.

GMAT Track of thought 2

Plugging in the values;
A. 3x/2 = 3*0/2 = 0 (even). Eliminate.
B. 3x/2 + 1 = 0 + 1 = 1 (Odd) Retain
C. 3\(x^2\) = 3*0 = 0 (Even) Eliminate
D. 3\(x^2 \)/ 2 = 0/2 = 0 (Even) Eliminate
E. 3\(x^2\) / 2 + 1 = 1 (Odd) Retain

GMAT Track of thought 3

So now I have reduced down to two options B & E. I can plug in multiple times as long as I plug in the right value.
The next simple value after 0 I can use is 2.
Using that, B is 3 * 2/2 = 3 & E is 3 * (22) /2 + 1= 7 (Even) and hence can be eliminated.
So the correct answer choice is E.

How would a CONCEPT ORIENTED & LOGICAL GMAT aspirant think through here?

GMAT Track of thought 1

Since I know what evens & Odds are and I know the Operations on them let me use those concepts and eliminate incorrect answer choices!

GMAT Track of thought 2

Even integers can be represented as 2n where n is an integer.

Using this in the answer choices which all options can I eliminate?
A. 3x/2 = 3 * 2n/2 = 3n .If n is odd then 3 being odd odd * odd will give me an Odd result but if n is even then odd * even will generate an even value. This option is NOT ALWAYS providing an Odd value. Eliminate.

B. 3x /2 + 1 = 3n + 1. Going by what I did in option A, I could have 3n Odd or Even and hence 3n + 1 can be Even or Odd respectively. This option is NOT ALWAYS providing an Odd value too. Eliminate.

C. 3\(x^2\) = 3* (2n)2 = 12n2 and this is always Even since 12 is a multiple of 2.Eliminate.

D. 3\(x^2\) /2 = 6n2 and this is always Even since 12 is a multiple of 2.Eliminate and Mark E as the correct answer choice.

E. 3\(x^2\) /2 + 1 = 6n2 + 1 = Even + 1 = Odd. This answer choice will ALWAYS return an Odd value and hence is the correct answer choice.
Hope this post of mine has added value to your learning.

PS- You can be a lethal hybrid mix of both logic, concepts & smartness as long as you keep practicing "thinking through" on the questions and the answer choices with every encounter you make with the Official Questions.

Devmitra Sen
GMAT Mentor

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