x is the sum of y consecutive integers. w is the sum of z

Question

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT 
 
A. x = w 
B. x > w 
C. x/y is an integer 
D. w/z is an integer
E. x/z is an integer

sriharimurthy · Accepted Answer

For a set of 'n' consecutive integers, the sum of integers is : 
(a) always a multiple of 'n' when n is ODD
(b) never a multiple of 'n' when n is EVEN
(Note : Try it out!)

From y = 2z, we can conclude that x is the sum of consecutive integers for an EVEN number of terms since y will always be even.

Thus x/y can never be an integer.

Answer : C

Bunuel · Answer

• If  k  is odd, the sum of  k  consecutive integers is always divisible by  k . Given  \{9,10,11\} , we have  k=3  consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If  k  is even, the sum of  k  consecutive integers is never divisible by  k . Given  \{9,10,11,12\} , we have  k=4  consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of  k  consecutive integers is always divisible by  k! , so by  k  too. Given  k=4  consecutive integers:  \{3,4,5,6\} . The product of 3*4*5*6 is 360, which is divisible by 4!=24.

h2polo · Answer

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

Case 1: x = w 
if y =   z =  
5=5, TRUE

Case 2: x > w 
if y =   z =  
5>1, TRUE

Case 3: x/y is an integer 
if y =   x = 9
9/3 is an integer, TRUE

Case 4: w/z is an integer
if z =   w = 2
2/1 is an integer, TRUE

Case 5: x/z is an integer
if x = 9, z =  
9/1 is an integer, TRUE

Am I doing something wrong here?

kp1811 · Answer

C

a) x= w if x = (1+2+3+4+5+6) = 21 and w = (6+ 7+8)=21. 
b) x>w if x= (1+2+3+4+5+6) = 21 and w = (4+5+6)= 15
c) x/y => x = (1+2+3+4+5+6) = 21 and y = 6 then x/y is NOT an integer 
=> x = (1+2+3+4) and y = 4 then x/y is NOT an integer
d) w/z =>w = (6+ 7+8)=21 and z=3 then w/z = 3
e) x/z => x = (1+2+3+4+5+6) = 21 and z = 3 then x/z = 7

xcusemeplz2009 · Answer

IMO (C).

y will always be even since y=2z

assume 6 numbers 
n-2 n-1 n n+1 n+2 n+3
x=6n+3, y=6, z=3

x/z = 2n+1

x/y = n+1/2 ==> cannot be integer

hrish88 · Answer

OA is C

sum of even no is not always odd.

consider 1+2+3+4=10 which is even

For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms .

consider above example 10 is not multiple of 4.

But For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.

1+2+3=6 multiple of 3

jeeteshsingh · Answer

since y = 2z... it is even...

x is the sum of y (even no) of integers....

Hence x/y would never be an integer....

Therefore C....

Note 
  For any set of consecutive integers with an even number of terms, the sum of the integers is never a multiple of the number of terms. 
 But For any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.

GyanOne · Answer

The question can be restated as: 
If x is the sum of y consecutive integers, and w is the sum of y/2 consecutive integers, and neither of y or z is 0, then which of the following can NEVER be true?

Note that since y>0 and z=y/2 and z>0 and both y and z are integers, y will always be an even integer.

(a) x=w. This is certainly possible. Take x=-2,-1,0,1,2,3 and w=0,1,2
(b) x>w. This is certainly possible. Take x=-2,-1,0,1,2,3 and w=-2,-1,0
(c) Correct answer. You can never make a case that satisfies this. Why? 
Sum of the AP = (y/2)   
If y = 2k as y is always even, Sum = x = k  
If we divide this by y, i.e. 2k, we get x/y = k /2k. This is an odd number divided by an even number, and so can never be an integer. 
(d) This is certainly possible. Evaluates to  /2, which can be true if z is odd. 
(e) This is certainly possible. Take x=-2,-1,0,1 and then z=2, and x/z=-1, which is an integer.

KarishmaB · Answer

This is how you can work it out:

x = sum of y con. integers
w = sum of z con. integers

Plug in values such that most of these are easily made true.
Since z is in the denominator, I will try to put z = 1 to get w/z and x/z as integers without any complications.

If z = 1, y = 2
w = any one integer
x = sum of 2 consecutive integers
w/z and x/z will be integers so d and e can be true.

x = w is obviously possible. Say w = 9 and x = 4+5
x > w is also possible say if w = 8 while x = 4+5
Hence both a and b can be true.

Answer must be c.

vidyad · Answer

x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

x = w 
x > w 
x/y is an integer 
w/z is an integer
 x/z is an integer

average1= x/y
average2=w/z

y=2z,

implies that for every value of z y is even.
ie.
z=even say 4 
y=2*4 =6 (even)

z=odd say 3

y=2*3 =6 (even)

the average of even numbers is not integer , so the answer is C.

ChiranjeevSingh · Answer

For these kind of questions, we need to parse through the options to find an option which can NEVER be true. There are three scenarios possible with any option:
1. We can easily find out that this is possible. Then, we can eliminate this option and move forward.
2. We can easily find out that the option is never true. Thus, we can stop here because this is the answer.
3. Nothing can be easily inferred. We need to plug in values and do some reasoning to figure out. These options can be skipped for the moment to come back later, if we don't find an answer among other options. We want to SAVE our time.

Here is my attempt at this:
A. x = w - just looking at it, i can't say if this is possible or not. To find so will take time.I skip it to come back later
B. x > w - Easily possible is x is the sum of first 2 integers and w is the sum of first 1 integer
C. x/y is an integer - come back later
D. w/z is an integer - easily possible if z=1, w=1
E. x/z is an integer  - easily possible if z=1, x=3

now, we are left with A and C options. I begin with option C.

x=sum of consecutive y (or 2z) integers = n + (n+1) + (n+2) +....+ (n+2z-1) = 2zn + (1+2+3+...+2z-1) = 2zn + 2z*(2z-1)/2 (using formula for sum of first n natural numbers)
=> x/y or x/2z = n + (2z-1)/2
we know n is an integer. The other component can never be an integer, since the numerator is always odd and denominator is 2. So, x/y cannot be an integer.

Thus, C is the answer.

Cheers,
CJ

streetking · Answer

thanks this is a brilliant explanation.

where could i get other number properties such as this?

Bunuel · Answer

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 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

plpurva · Answer

The sum of an  odd  number of consecutive integers is  always  a multiple of the number of integers.
The sum of an  even  number of consecutive integers is  never  a multiple of the number of integers.

x is the sum of y consecutive integers. w is the sum of z

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

x is the sum of y consecutive integers. w is the sum of z

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards