jlgdr wrote:
The average (arithmetic mean) of the four distinct positive integers p, q, s and t is x. If p and q are odd, and s and t are even, which of the following must be true?
I. x is an integer
II. (x−0.5) is an integer.
III. 2x is an odd integer.
A. I only
B. II only
C. III only
D. II and III only
E. None of the above
Using the information given to us:
p = 2a+1
q = 2b+1
s = 2c
t = 2d
\(x = \frac{p+q+s+t}{4} = \frac{2a + 2b + 2c + 2d + 2}{4} = \frac{2(a+b+c+d) + 2}{4} = \frac{a+b+c+d+1}{2}\)
Now, whether x is even or odd depends on the values of a, b, c and d. We have no restrictions on these as in they can be even or odd. If the sum (a+b+c+d) is odd, the numerator will become even and x will be an integer. If the sum (a+b+c+d) is even, then the numerator will become odd and x will be 0.5 more than an integer.
Hence, neither I nor II is essential.
III is not essential either since 2x will be odd only when the sum (a+b+c+d) is even.
So none of the three MUST be true.
Answer (E)
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Karishma
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