Question

The product of $n$ positive integers is $1$, then their sum is a positive integer, that is

• A. equal to $1$
• B. equal to $n+n^2$
• C. divisible by $n$
• D. never less than $n$

Answer 1

Answer: (D)

never less than $n$

Given $x_1.x_2.x_3...x_n=1$

Since $A.M.\ge G.M.$

$\therefore \left(\frac{x_1+x_2+...+x_n}{n}\right)\ge {(x_1.x_2...x_n)}^{1/n}={\left(1\right)}^{1/n}=1$

$\Rightarrow x_1+x_2+...+x_n\ge n$

Hence $x_1+x_2+...+x_n$ can never be less than $n$.