SOLUTION: If the product of the integers a, b, and c is 1, then what is the smallest possible value of (a+1)(b+1)(c+1)?

Question 783031: If the product of the integers a, b, and c is 1, then what is the smallest possible value of (a+1)(b+1)(c+1)?

Answer by tommyt3rd(5050) (Show Source): You can put this solution on YOUR website!
Expanding and simplifying...
(a+1)(b+1)(c+1)=
(a+1)(bc+b+c+1)=
abc+ab+ac+bc+a+b+c+1=
ab+bc+ac+a+b+c+2
Since their product is 1, the only possibilities are a,b,c={-1,1} (why?)
Furthermore, there must be an odd number of positives. To find the minimum we can ignore the case a=b=c=1 - that would be the maximum.
Case1
if say a=b=1, c=-1, then
ab+bc+ac+a+b+c+2=
1-1-1+1+1-1+2=
2
Case2
if say c=1, a=b=-1, then
ab+bc+ac+a+b+c+2=
1-1-1-1-1+1+2=
0
Minimum value is 0
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