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If x represents the product of the first 100 positive odd integers and

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If x represents the product of the first 100 positive odd integers and  [#permalink]

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New post 10 Jan 2018, 20:51
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A
B
C
D
E

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If x represents the product of the first 100 positive odd integers and y represents the sum of the first 100 positive odd integers, then the value of x−y is

A. Even
B. Odd
C. Negative
D. Divisible by 3
E. Divisible by 6

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Re: If x represents the product of the first 100 positive odd integers and  [#permalink]

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New post 10 Jan 2018, 21:05
Product of any nbr of odd integers will always be odd. So X is Odd. Sum of even nbr of odd integers will be always even so Y will be Even. So X-Y will be difference of Odd and Even which will be Odd. So answer is B

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Re: If x represents the product of the first 100 positive odd integers and  [#permalink]

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New post 10 Jan 2018, 21:11
B. Odd

The product i.e. X will be an odd number as it is 1.3.5.7..so on.
Similarly addition i.e Y can be easily calculated as 10,000 which is even.

(By pattern you can see that for Y, it is always n^2, where n is the number of first odd integers)

Hence the subtraction is an odd number.

Hence it is Odd - Even = Odd.
Option B.


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Re: If x represents the product of the first 100 positive odd integers and  [#permalink]

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New post 10 Jan 2018, 21:25
Bunuel wrote:
If x represents the product of the first 100 positive odd integers and y represents the sum of the first 100 positive odd integers, then the value of x−y is

A. Even
B. Odd
C. Negative
D. Divisible by 3
E. Divisible by 6



Normally choices help us even before we start..
we are looking at integers and it can be either odd or even, so answer has to be A or B..

\(x = 1*3*5.......*199\)... Product of odd integers will be ODD
\(y = 1+3+5.......+199\), there are 100 numbers of ONE type, so irrespective of all being ODD or all being even, the SUM will be EVEN
so x-y = Odd-Even = Odd..
B
But lets look at other choices..
A) even:
It cannot be EVEN
C) Negative :-
x is product and y sum so x >y, so x-y>0
D) divisible by 3..
product consists of multiple of 3, so x= 1*3*5..*199 will be div by 3
SUM.. sum of any three consecutive odd integers will be div by 3, so had there been 99 integers, it would have been div by 3, but there are 100 integers so SUM is NOT div by 3
so Multiple of 3-Non-multiple of 3 = Non-multiple of 3.. ans No
E) div by 6..
If not div by3, it will NOT be div by 6.. hence NO
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: If x represents the product of the first 100 positive odd integers and &nbs [#permalink] 10 Jan 2018, 21:25
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