catty2004 wrote:
From the consecutive integers -10 to 10 inclusive, 20 integers are randomly chosen with repetitions allowed. What is the least possible value of the product of the 20 integers?
A. (-10)^20
B. (-10)^10
C. 0
D. –(10)^19
E. –(10)^20
Each of these randomly selected integers could be a negative number, a positive number, or zero, so their product could be negative. Their product has the least possible value if it is a negative number with the greatest possible absolute value.
First, we select the number with the greatest possible absolute value for each place, except one. Then, we must be careful to select the right number for the last place because we have to consider not only its absolute value but also its sign.
Although -10 and 10 have the same absolute value, we should choose 10s for the first 19 places because negative numbers always complicate things. Since \(10^{19}\) is positive, we must select the negative number with the greatest possible absolute value for the last place in the product.
\(10^{19}(−10)=(−)10^{19}(10)=(−)10^{20}=−10^{20}\)
\(−10^{20}\) is the same number as \(−(10)^{20}\) in the correct answer choice.
Answer: E
If the given set of integers consisted of only non-negative or only non-positive numbers, then we would need different strategies.