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20 May 2020, 07:37
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E
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Difficulty:
75%
(hard)
Question Stats:
56% (02:21) correct
44%
(01:53)
wrong
based on 292
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How many integer values of x satisfy \((x + 2)(x + 4)(x + 6)...(x + 100) < 0\)?
A. 50
B. 49
C. 47
D. 26
E. 25
Are You Up For the Challenge: 700 Level Questions
A. 50
B. 49
C. 47
D. 26
E. 25
Are You Up For the Challenge: 700 Level Questions
20 May 2020, 08:02
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Bunuel
Number of terms in the equation = \(\frac{100-2}{2}\) + 1 = 50
Clearly, if the value of x is positive or 0, the LHS will be greater than 0, therefore, we must focus on negative integers.
Let's consider the largest negative integer = -1
x = -1 when added to the LHS will yield a positive LHS value, thus, does not satisfy the equation
x = -2 will yield LHS = 0 as (x+2) = (-2+2) = 0, does not satisfy the equation
x = -3, then one term, namely x+2 will be negative, all others will be positive. Thus, the product will be < 0
We got our first possible integer value of x.
Upon repeating this process for some other negative numbers, we'll be able to see a pattern in play:
Every fourth negative number after -3 will make the LHS negative and hence, satify the equation.
These integers are:
-3, -7, -11, ........., -99
To find the number of terms, you can view this as an AP with a = -3 and d = -4
-99 = -3 + (n - 1) * -4
\(\frac{-96}{-4} = n - 1\)
24 = n - 1
n = 25 (E)
General Discussion
20 May 2020, 08:20
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We want the product to be a negative number as the expression must be less than 0.
There are a total of 50 expressions and hence an odd number of negatives would only result in negative number. The range of such numbers will be [-3 to -99], inclusive. How I got to this range is by trying. Put in -1, you have all the numbers as positive and hence the product won't be less than 0. All the even negative integers won't be considered as they will make one of the expression as a 0. Moving onto -5, we will again get two negative expressions resulting to a positive product. Now we move onto -7 which will result in 3 negative numbers getting multiplied.
So the result will have {-3,-7,-11,-15.......-99} which are 25 in number and hence:
Answer - E
There are a total of 50 expressions and hence an odd number of negatives would only result in negative number. The range of such numbers will be [-3 to -99], inclusive. How I got to this range is by trying. Put in -1, you have all the numbers as positive and hence the product won't be less than 0. All the even negative integers won't be considered as they will make one of the expression as a 0. Moving onto -5, we will again get two negative expressions resulting to a positive product. Now we move onto -7 which will result in 3 negative numbers getting multiplied.
So the result will have {-3,-7,-11,-15.......-99} which are 25 in number and hence:
Answer - E
