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For integers, $a$, $b$ and $c$, what would be the minimum and maximum values respectively of $a+b+c$  if $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =0$?

  1. $\text{-3 and 3}$
  2. $\text{-1 and 1}$
  3. $\text{-1 and 3}$
  4. $\text{1 and 3}$
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Best answer
Given that $a,b$ and $c$ are integers.

and $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =0$

we know that $\log \mid a \mid + \log \mid b \mid + \log \mid c \mid =\log(\mid a \mid\cdot\mid b\mid\cdot\mid c\mid)$

$\implies\log(\mid a \mid\cdot\mid b\mid\cdot\mid c\mid)= 0$

$\implies \mid a \mid\cdot\mid b\mid\cdot\mid c\mid={10}^{0}$

$\implies \mid a \mid\cdot\mid b\mid\cdot\mid c\mid=1$

For minimum value of $a+b+c$ we can put $a=b=c=-1$ and get
$(a+b+c)_{\min}=-1-1-1=-3$

For maximum value of $a+b+c$ we can put $a=b=c=1$ and get
$(a+b+c)_{\max}=1+1+1=3$

So, (A) is the correct answer.
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