Which of the following describes the set of all possible values of the

Question

Which of the following describes the set of all possible values of the positive integer k such that, for each positive odd integer n, the value of n/k is midway between consecutive integers?

A. All positive integers greater than 2
B. All prime numbers
C. All positive even integers
D. All even prime numbers
E. All positive even multiples of 5

PS55471.01

ScottTargetTestPrep · Accepted Answer

We see that n can be 1, 3, 5, and so on. If k = 2, then 1/2 is midway between 0 and 1, 3/2 is midway between 1 and 2, 5/2 is midway between 2 and 3, and so on. Since any other value of k would not have such a property, k must be 2. Note that 2 is the only even prime number, so choice D is correct.

Answer: D

Kanika3agg · Answer

n/k should be such that it lies midway between consecutive integers.
Best way to solve a question - imagine what it is trying to say.

Consecutive integers are 1,2,3...
midway would be 1.5, 2.5 etc...
Odd numbers divided by what k would give us this value? - 2 which is all even prime numbers.

zhanbo · Answer

I have to wonder if "midway between consecutive integers" is a standard math term? Thanks!

Karmesh · Answer

There are 2 things you need to catch here
1. the set of all possible values of the positive integer k
2. the value of n/k is midway between consecutive integers

When will the value of n/k always be midway between something? or let's rephrase it, midway= half so, basically when will n/k be half? When k=2. Seems pretty straightforward now doesn't it

Now, just look for an option that gives you k=2

Brucewayne00726 · Answer

Hi,

Can anyone please explain this question in more simpler terms?

khan0210 · Answer

Which of the following describes the set of all possible values of the positive integer k such that, for each positive odd integer n, the value of n/k is midway between consecutive integers?

There are three parts to this question:

1) Set of all possible values of positive integer K

:. k>0 and integer - so k=1 k=2 k=3 etc.

2) Each positive odd integer n
:. n= 1, 3, 5, ...

3) the value of n/k is midway between consecutive integers
consecutive integers 0,1,2,3,4,...
:.n/k to be midway means n/k would have to be 1/2, 1.5, 2.5 which are midpoints of consecutive integers.

The only value of K that will fulfill condition 3 is if k=2. 
:. if n=1 n/k= 1/2; n=3 n/k=3/2 i.e 1.5 and so on

And 2 is the only even prime number. Hence, D.

Hope this helps.

Cellchat · Answer

What about, for example, if n = 7 and k = 14, then the result of n/k is still midway between consecutive integers (0 and 1).

k does not have to be 2. It could be 2, 6, 10, 14, 18, 22...

Any help?

renzofig · Answer

Hi  Cellchat , the problem is that the question actually states   "for each positive odd integer n"  . That means, that regardless of the value of n (as long as it is a positive odd integer), the value will always be halfway between 2 consecutive integers.

For example, lets say n=5 and k=14. 5/14 is not midway between 2 integers. Btw, "midway" means exactly in the middle. For example: 2.5, 3.5 or 4.5 are midway between 2 integers, whereas 4.2 is not.

Hope this helps.

queenztea · Answer

How could C not be the answer? 
It also given 0.5 everytime.

Basshead · Answer

\frac{n}{k}

let  n = 3 
let  k = 8

\frac{3}{8}  is not midway between consecutive integers. The only value for k that would give us exactly the midway is 2.

3/ 2  = 1.5.

Rahulbasu007 · Answer

Takeaways:
0 is an integer.
1 is an integer, and so on...
Midway between consecutive integers: The number should be divided by 2.
Examples: 
0.5, (midway between 0 and 1) -> 1/2
1.5 (midway between 2 and 3) -> 3/2
2.5 (midway between 2 and 3) -> 5/2
and so on. 
Note that any other odd number or even number is not fitting the criteria.

2 is an even number.
2 is a prime number.

Therefore, option D.
I got confused with the term "midway" between two consecutive numbers, initially.

Kinshook · Answer

Asked: Which of the following describes the set of all possible values of the positive integer k such that, for each positive odd integer n, the value of n/k is midway between consecutive integers?

Only k = 2 is possible.
e.g. n=3; k = 2; n/k = 3/2 =1.5 is midway between consecutive integers

A. All positive integers greater than 2
B. All prime numbers
C. All positive even integers
D. All even prime numbers; k = 2 is the only case
E. All positive even multiples of 5

IMO D

Saupayan · Answer

This testcase doesn't help eliminate C, because 2 is also a positive even number.

satish_sahoo · Answer

Values for n = 1,3,5,...so on..

Imagine this on a number line. Midway point of 0 to 1 is  1/2 , similarly if you notice midway point of 1 to 2 is  3/2 , and again mid point  5/2  for 2 to 3.

We can only have 2 as the value of k, which is the only even prime number.

Hence, D

sachi-in · Answer

The problem is almost 90% solved when we consider n = 2a+1 ( because n is odd )

and k as some random number

When we do  n/k  =  ( 2a + 1 )/k  =  2a/k  +  1/k

Looking at above result it's very easy to figure out K = 2 :

Since 2a/k is an integer  and if this  number ( n/k ) is midway between two number that means 1/k  =  1/2  or  0.5

|----------|---------|
I ----- I+0.5 ----- I+1

WinWinChickenDin · Answer

Can anyone recommend similar problems I could practice? I had difficulty translating what this problem was asking me to find and thus could not figure out what the solution was.

Bunuel · Answer

This question belongs to the broad category of Number Properties, which you can filter by applying the respective Category filters in the Problem Solving forum : https://gmatclub.com/forum/download/file.php?id=83726 For theory check the below links. 2. Properties of Integers Theory Math: Number Theory Number Properties: Tips and hints Advanced Number Properties on GMAT (All 5 parts) Do you make these 3 mistakes in GMAT Even-Odd Questions? Everything about Factorials on the GMAT Solving Complex Problems Using Number Line Questions Tough New Set: Number Properties Set Power of a Number in a Factorial Problems Number Properties Ability Quiz Special Numbers and Sequences Problems The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! Trailing Zeros Problems PS Questions For more check Ultimate GMAT Quantitative Megathread Hope it helps. GMAT-Club-Forum-3oscnn9j.png

Usernamevisible · Answer

Keep reading the solution till the end even if you dont understand 
 Feels difficult - dont invest lot of time, ok to let go, so if you have completed reading it once, thats' it.

The condition says:
For every positive odd integer n, the value of n/k is midway between consecutive integers.
Translate:
n/k = a number ending in .5
for every odd n.
----------------

Start with the smallest odd integer:
n = 1
Then:
1/k
must be midway between consecutive integers.
--------------------------------------------

What positive numbers are midway between consecutive integers?
0.5, 1.5, 2.5, 3.5, ...
But:
1/k ≤ 1
because k is a positive integer.
The only halfway value that is positive and not greater than 1 is:
0.5
Therefore:
1/k = 0.5
k = 2
-----

Check that k = 2 works.
Odd numbers:
1, 3, 5, 7, 9, ...
Divide by 2:
1/2 = 0.5
3/2 = 1.5
5/2 = 2.5
7/2 = 3.5
9/2 = 4.5
Every result ends in .5, so every result is midway between consecutive integers.
Therefore k = 2 works.
----------------------

Why can't any other k work?
Try k = 4:
1/4 = 0.25
Not midway between consecutive integers.
Fails immediately.
Try k = 6:
1/6 ≈ 0.167
Not midway between consecutive integers.
Fails immediately.
Since the condition must hold for every odd n, checking n = 1 alone forces:
k = 2
-----

Answer:
The set of all possible values of k is {2}.
The only choice that represents {2} is:
D. All even prime numbers
because 2 is the only even prime.

---------------------------------

GMAT takeaway:
Whenever a question says "for every odd integer n," try the smallest odd integer first:
n = 1
Very often, that single test value collapses the entire problem.

Which of the following describes the set of all possible values of the

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