ziyuenlau wrote:If a + 1 = (20/a) and b is the average of a set of c consecutive integers, where c is odd, which of the following must be true?
I. (a^2)(b^2)(c^2) is even.
II. a + b + c is odd.
III. ab[(c^2) + c] is even.
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
Source : Manhattan Prep
Hi ziyuenlau,
We have a + 1 = (20/a)
=> a^2 + a = 20
=> a^2 + a - 20 = 0
=> a^2 + 5a - 4a - 20 = 0
=> a(a + 5) - 4(a +5) = 0
=> a = -5 or 4 => a can be even or odd
We know that c is odd.
Let us find out the nature of b.
Say there are three consecutive integers, c = 3.
Case 1: Set: {2, 3, 4} => Average = b = 3 (odd)
Case 2: Set: {3, 4, 5} => Average = b = 4 (even)
b can be even or odd.
Let's see each statement one by one.
S1: (a^2)(b^2)(c^2) is even.
Say a and b each is odd
(a^2)(b^2)(c^2) = (Odd^2)(Odd^2)(Odd^2) = Odd. It's not a must be true statement.
S2: a + b + c is odd.
Say a and b each is odd.
=> a + b + c = Odd + Odd + Odd = Odd. It's not a must be true statement.
S3: ab[(c^2) + c] is even.
ab[(c^2) + c] = ab[Odd^2 + Odd] = ab[Odd + Odd] = ab[Even] = Even.
=> Whatever be the nature of a and b, ab[(c^2) + c] is always even.
Statement III must be true.
The correct answer:
C
Hope this helps!
Relevant book:
Manhattan Review GMAT Math Essentials Guide
-Jay
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