A positive integer which when added to 1000 gives a sum which is greater than when it is multiplied by 1000. That positive integer is: a) 1 b) 5 c) 7 d) 3

Let's break down the problem step by step:

We are asked to find a positive integer \( x \) such that when it is added to 1000, the sum is greater than when it is multiplied by 1000. In other words, we are given the condition:

x + 1000 > 1000 * x

### Step 1: Set up the inequality
We can write the given condition as:

x + 1000 > 1000 * x

### Step 2: Rearrange the inequality
To solve for \( x \), we will move all terms involving \( x \) to one side of the inequality. Subtract \( x \) from both sides:

1000 > 1000 * x - x

Factor out \( x \) on the right side:

1000 > x * (1000 - 1)

Simplify:

1000 > x * 999

### Step 3: Solve for \( x \)
Now, divide both sides of the inequality by 999:

1000 / 999 > x

This simplifies to approximately:

1.001 > x

### Step 4: Interpret the result
Since \( x \) must be a positive integer, the only integer less than 1.001 is 1. Therefore, \( x = 1 \) is the only solution that satisfies the inequality.

### Step 5: Verify the solution
Let’s verify by substituting \( x = 1 \) into the original condition:
- Adding: 1 + 1000 = 1001
- Multiplying: 1 * 1000 = 1000

Indeed, 1001 is greater than 1000, so \( x = 1 \) satisfies the condition.

### Final Answer:
The positive integer is 1. Therefore, the correct option is:
a) 1