For the positive integers a, b, c find the minimum value of `(-1)^a + (-1)^b + (-1)^c + (-1)^d`, If a + b + c + d = 1947

Options

A. A)-1
B. B)-2
C. C)-3
D. D)-4

Answer

To find the minimum value of the expression \((-1)^a + (-1)^b + (-1)^c + (-1)^d\) given that \(a + b + c + d = 1947\) and \(a, b, c, d\) are positive integers, we can follow these steps:

### Step 1: Understand the expression
The expression \((-1)^x\) yields:
- \(1\) if \(x\) is even
- \(-1\) if \(x\) is odd

### Step 2: Analyze the sum of \(a, b, c, d\)
Since \(a + b + c + d = 1947\), we need to consider how many of these integers can be odd or even to minimize the expression.

### Step 3: Determine the parity of the integers
To minimize the expression, we want to maximize the number of \(-1\)s. This occurs when we have as many odd integers as possible.

### Step 4: Assign values to \(a, b, c, d\)
Let's set three of the integers \(a, b, c\) to be odd and the fourth integer \(d\) to be even. The simplest odd integers we can use are \(1\):
- Let \(a = 1\)
- Let \(b = 1\)
- Let \(c = 1\)

Now we can find \(d\):
\[ d = 1947 - (a + b + c) = 1947 - (1 + 1 + 1) = 1944 \]
Here, \(d\) is even.

### Step 5: Calculate the expression
Now substituting the values into the expression:
\[ (-1)^a + (-1)^b + (-1)^c + (-1)^d = (-1)^1 + (-1)^1 + (-1)^1 + (-1)^{1944} \]
Calculating each term:
- \((-1)^1 = -1\)
- \((-1)^1 = -1\)
- \((-1)^1 = -1\)
- \((-1)^{1944} = 1\) (since 1944 is even)

So, we have:
\[ -1 - 1 - 1 + 1 = -2 \]

### Conclusion
Thus, the minimum value of the expression \((-1)^a + (-1)^b + (-1)^c + (-1)^d\) is \(-2\).