x is the sum of y consecutive integers

[Guest]

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

1. x = w
2. x > w
3. x/y is an integer
4. w/z is an integer
5. x/z is an integer

Explanation given:
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. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. 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. For example, the sum of 1, 2, 3, and 4 (four consecutives -- an even number) is 10, which is not a multiple of 4.

The question tells us that y = 2z, which allows us to deduce that y is even. Since y is even, then the sum of y integers, x, cannot be a multiple of y. Therefore, x/y cannot be an integer; choice C is the correct answer.


Question:
Can you please explain another way to solve this problem? Is there another way to do it instead of plugging in numbers (which would take a long time in this case)?

[Guest]

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

1. x = w
2. x > w
3. x/y is an integer
4. w/z is an integer
5. x/z is an integer

Explanation given:
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. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. 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. For example, the sum of 1, 2, 3, and 4 (four consecutives -- an even number) is 10, which is not a multiple of 4.

The question tells us that y = 2z, which allows us to deduce that y is even. Since y is even, then the sum of y integers, x, cannot be a multiple of y. Therefore, x/y cannot be an integer; choice C is the correct answer.

Question: Can you please explain another way to solve this problem? Is there another way to do it instead of plugging in numbers (which would take a long time in this case)?

y = 2z
if z = 1, y 2
if z = 2, y = 4
if z = 3, y = 6
if z = 4, y = 8

so y is always even and average of even consecutive integers is always not an integer.

lets write down some even consecutive integers. suppose y = 4:

so the integers are: a, a+1, a+2, a+3
x = a + a+1 + a+2 + a+3
x = 4a + 6
so 4a+6 is not evenly divisible by 4. therefore x/y is not an integer...

[UPA]

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

1. x = w
2. x > w
3. x/y is an integer
4. w/z is an integer
5. x/z is an integer

Explanation given:
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. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. 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. For example, the sum of 1, 2, 3, and 4 (four consecutives -- an even number) is 10, which is not a multiple of 4.

The question tells us that y = 2z, which allows us to deduce that y is even. Since y is even, then the sum of y integers, x, cannot be a multiple of y. Therefore, x/y cannot be an integer; choice C is the correct answer.

Question: Can you please explain another way to solve this problem? Is there another way to do it instead of plugging in numbers (which would take a long time in this case)?

y = 2z
if z = 1, y 2
if z = 2, y = 4
if z = 3, y = 6
if z = 4, y = 8

so y is always even and average of even consecutive integers is always not an integer.

lets write down some even consecutive integers. suppose y = 4:

so the integers are: a, a+1, a+2, a+3
x = a + a+1 + a+2 + a+3
x = 4a + 6
so 4a+6 is not evenly divisible by 4. therefore x/y is not an integer...

wow i forgot to put my username.

[rfernandez]

UPA's solution is akin to how I would solve it as well. I usually find it helpful to express consecutive sets using variables like a, a+1, a+2, etc. and seeing what insights I can draw from that analysis.

Rey

[viv09]

average of even consecutive integers is always not an integer.

Is this a rule?

eg.
2+4+6/3 = 12/3 = 4

Can some one please explain the statement given above.

Thanks

[saptadeepc]

average of even consecutive integers is always not an integer.

Is this a rule?

eg.
2+4+6/3 = 12/3 = 4

Can some one please explain the statement given above.

Thanks

what he meant is

average of even "numbers" of consecutive numbers is not an interger

for example
1 + 2 / 2 OR 1 + 2 + 3 + 4 / 4 OR 1 + 2 + 3 + 4 + 5 + 6 / 6

are never integers

[viv09]

oh, I think I got caught up in the words... Thanks!!

[jnelson0612]

Great! :-)

[subrat308]

Another approch:

X is the sum of y integers.

X=Y/2{2a1+(Y-1)*1}------a1 is here the 1st no of the series and diff is 1 as consecutive integers.

So X=Y/2{2a1+2Z-1}---as Y=2Z

So X/Y={2(a1+Z)-1}/2----This can never be an integer as (2k-1)/2 will not give you integer.

Option A) Possible

B)Possible

C)Not possible as shown above.

D)W=Z/2{2a2+(Z-1)*1}------a2 is here the 1st no of the series and diff is 1 as consecutive integers.

So W/Z=1/2{2a2+Z-1}--- It may be possible sometime(Ex 4,5,6)

E)So X=Y/2{2a1+2Z-1}---as Y=2Z

So X/Z={2(a1+Z)-1}---- Always an integer So Possible

[jnelson0612]

Wow! I have seen that equation before and that's a nice use of it. Thanks for chiming in!

On this one, I personally like to plug numbers. If you use very small numbers you can manipulate them fairly easily.

[tathiane.thompson]

xx

[RonPurewal]

Hello,

I am really struggling with this problem.
I get that the number y must be even because of the 2, but what I don't understand is why are you assuming that the number of terms is also even?

"y" IS the number of terms, according to this statement:
x is the sum of y consecutive integers

if that isn't clear, then put a number in the place of "y" and it should become more clear:
x is the sum of 8 consecutive integers --> this means that there are eight integers.

Please help I would appreciate some help. Also and what part is this concept on the Manhattan books?

the relevant concepts would be evens/odds, consecutive integers, and averages. but, as in the case of many other number-property problems, they've woven relatively simple concepts into a clever (and fiendish) problem.
so, it depends on what you are asking here.

* if you are asking, "is there a rule/formula that will just solve this?", then, no -- and that's the whole point of the math section of this exam! you can't just memorize rules; you have to manipulate simpler concepts to figure out more complex or specialized truths.

* if you are asking, "where can i find the basic concepts that are relevant here?", then see above.

[tathiane.thompson]

Thank you very much.
I have been out of high school for over 10 years and I feel very rusty.
I am taking the test this Saturday and I confess and I am very nervous but at the same time, I am trying my best.

Thank you.

[jlucero]

If it's on the GMAT, it's designed to be tricky for some people. You (and 99.99% of everyone else) are going to get some questions wrong on the test. Just do your best on each question and try your best not to get flustered on the problems you don't know.

Oh, and good luck!

[adm45]

Can you explain without plugging in numbers, why Z is odd or even? When I saw Y= 2Z, I thought that Y is even. So Even divided by 2 equals Z. Z equals even number.


Also, would plugging in number be good use here? if so, please demonstrate.

Is this a concept that must be memorized: "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. For example, the sum of 1, 2, and 3 (three consecutives -- an odd number) is 6, which is a multiple of 3. 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."?

Thanks,