Q. 6 For integers a, b and c, what would be the minimum and maximum values
respectively of a + b + c if log |a| + log |b| + log |c| = 0?
Consider the equation:
log |a| + log |b| + log |c| = 0
where a, b, c are non-zero integers. We are asked to find the
minimum and maximum values of a + b + c.
Simplifying the Logarithmic Expression
Using logarithmic properties:
log |a| + log |b| + log |c| = log (|a| · |b| · |c|)
So the given equation becomes:
log (|a| · |b| · |c|) = 0
Since log 1 = 0, we get:
|a| · |b| · |c| = 1
Restricting Integer Values
The only integers whose absolute value equals 1 are:
- 1
- −1
Hence, each of a, b, c must be either 1 or −1.
Finding Minimum and Maximum of a + b + c
Possible extreme cases:
- Minimum: a = b = c = −1
a + b + c = −1 − 1 − 1 = −3 - Maximum: a = b = c = 1
a + b + c = 1 + 1 + 1 = 3
Correct Answer: Option (A) −3 and 3
Why Other Options Fail
| Option | Range Given | Reason It Fails |
|---|---|---|
| (B) | −1 and 1 | Too narrow; ignores all-positive and all-negative cases |
| (C) | −1 and 3 | Misses minimum value −3 |
| (D) | 1 and 3 | Excludes negative integer combinations |
Final Conclusion
Since a, b, c ∈ {−1, 1}, the sum a + b + c can only vary from:
−3 ≤ a + b + c ≤ 3
Final Answer: −3 and 3
