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Bunuel
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If y is the sum of x consecutive positive integers, and x>3, is y odd? 17 Mar 2021, 01:30
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E
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85% (hard)

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34% (01:42) correct 66% (02:00) wrong based on 91 sessions

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If y is the sum of x consecutive positive integers, and x>3, is y odd?

(1) x is odd.
(2) The largest number among the x consecutive integers is odd.


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E
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If y is the sum of x consecutive positive integers, and x>3, is y odd? 17 Mar 2021, 04:56
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Statement 1: X is odd

Case 1:

x = 3
first integer is a = 1

so y= a+a+1+a+2 = 3a+3= 6 = Even.

Case 2:

x = 3
First integer is 2

So Y= 3(2) +3 = 9 = Odd.

Statement 1 is insufficient.


Statement 2: The largest number among the x consecutive integers is odd.

Case 1: Largest Integer= 3

y = 1+2+3 = 6, Even.

Case 2 : Largest Integer = 5

y = 1+2+3+4+5 = 15, Odd.

Insufficient.

Combining 1 and 2 we get

Largest term is odd and the # of terms in series are also odd and we still cannot eliminate any of the above cases. Hence, answer should be E, in my opinion.
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If y is the sum of x consecutive positive integers, and x>3, is y odd? 17 Mar 2021, 05:54
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Bunuel
If y is the sum of x consecutive positive integers, and x>3, is y odd?

(1) x is odd.
(2) The largest number among the x consecutive integers is odd.


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If we take a pair of consecutive integers, their sum is odd. So if there are even numbers of pair, the sum will be even, and if there odd number of pairs, the sum will be odd.

(1) x is odd.
So the number of consecutive numbers are (certain pairs+e/o)
The bold portion above tells us that we cannot be sure about y.
Example: 1+2+3=6, which is even, while 2+3+4=9, which is odd
Insufficient

(2) The largest number among the x consecutive integers is odd.
We do not know how many integers are there.
1+2+3=6, while 1+2+3+4+5=15
Insufficient

Combined
We have one more integer than certain pairs of (odd+even).
And that number is Odd.
So if even pairs are there=> even*(odd+even)+odd=even*odd+odd=odd
But, if odd number of pairs are there=> odd*(odd+even)+odd=odd*odd+odd=Even
Insufficient

E
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If y is the sum of x consecutive positive integers, and x>3, is y odd? 17 Mar 2021, 12:32
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E .
If y is the sum of x consecutive positive integers, and x>3, is y odd?

(1) x is odd.

Sum will be even or odd depending on the the last number . .. Not sufficient

(2) The largest number among the x consecutive integers is odd.

Sum will be even or odd .. depending on x= even pairs or odd pairs .. or odd .. .. Not sufficient

Combining 1 and 2 also
Sum will be even or odd .. depending on x= even pairs or odd pairs .. or odd .. .. Not sufficient
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If y is the sum of x consecutive positive integers, and x>3, is y odd? 17 Mar 2021, 23:59
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If y is the sum of x consecutive positive integers, and x>3, is y odd?

(1) x is odd.
y = 1+2+3+4+5 = 15
y = 2+3+4+5 = 14
Not sufficient

(2) The largest number among the x consecutive integers is odd.
y = 1+2+3+4+5 = 15
y = 2+3+4+5 = 14
Not sufficient

(1) + (2)
y = 1+2+3+4+5 = 15
y = 2+3+4+5 = 14
Not sufficient

Option E

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If y is the sum of x consecutive positive integers, and x>3, is y odd? 18 Mar 2021, 01:15
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Bunuel
If y is the sum of x consecutive positive integers, and x>3, is y odd?

(1) x is odd.
(2) The largest number among the x consecutive integers is odd.


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If y is the sum of x consecutive positive integers, and x>3, is y odd?

The sum of x consecutive integers equals the mean multiplied by the number of terms, thus x*{mean}=y. Recall that in evenly spaced set mean=median, thus we have that x*{mean}=x*{median}=y.

The question asks whether y is odd.

(1) x is odd --> x*{median}=odd*{median}=y. If a set has odd number of terms the median is the middle term, when arranged in ascending/descending order, thus the median of the set must be an integer. For y to be odd the median must be odd, but the median of odd number of consecutive integers can be odd as well as even, for example, consider {1, 2, 3, 4, 5} and {1, 2, 3, 4, 5, 6, 7}. Not sufficient.

(2) The largest number among the x consecutive integers is odd. If the set is {0, 1, 2, 3}, then y is even but if the set is {1, 2, 3, 4, 5}, then y is odd. Not sufficient.

(1)+(2) The sets we considered to prove insufficiency of the first statement are still valid because both of them have the largest term odd (to satisfy the second statement). Thus even taken together the statements are not sufficient.

Answer: E.
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