A group of cats got together and decided to kill, between them, a total of 999919 rats. Every cat killed the same number of rats, and each cat killed more rats than the total number of cats. How many rats did each cat kill?

Introduction / Context:
This puzzle question tests your understanding of basic number theory, factors of an integer, and simple logical reasoning. We are told that a certain number of cats together killed exactly 999919 rats, that each cat killed the same number of rats, and that each cat killed more rats than there were cats in the group. The task is to determine how many rats each cat killed from the given options.

Given Data / Assumptions:

• Total number of rats killed altogether = 999919.
• There are n cats, where n is a positive integer.
• Each cat kills exactly k rats, where k is a positive integer.
• All cats kill the same number of rats, so total rats = n * k.
• Each cat kills more rats than the total number of cats, so k > n.
• We assume the given answer choices contain the correct value of k.

Concept / Approach:
The key idea is to treat this as a factorisation puzzle. Since n * k = 999919, both n and k must be positive integer factors of 999919. The condition that each cat kills more rats than there are cats (k > n) will help us decide which factor is k and which is n. We can use the options as possible values of k, and check which one divides 999919 exactly and also satisfies k > n.

Step-by-Step Solution:
Step 1: Let the number of cats be n and the number of rats killed by each cat be k.Step 2: From the statement, we have n * k = 999919.Step 3: Test the answer options as possible values of k: 991, 1000, 1009 and 2000.Step 4: Check divisibility: 999919 / 1000 is not an integer and 999919 / 2000 is not an integer, so 1000 and 2000 cannot be k.Step 5: Compute 999919 / 991 = 1009 (an integer), and 999919 / 1009 = 991 (also an integer). So the factor pairs are (n, k) = (991, 1009) or (1009, 991).Step 6: Apply the condition k > n. Only the pair n = 991, k = 1009 satisfies k > n, so each cat must have killed 1009 rats.

Verification / Alternative check:
You can quickly verify the answer by multiplying 991 * 1009. Using arithmetic or a calculator, 991 * 1009 = 999919, which matches the total number of rats given in the problem. Also check the inequality: 1009 > 991, so each cat killed more rats than the number of cats. This confirms that k = 1009 is consistent with all conditions.

Why Other Options Are Wrong:
Option 991 is wrong because if k = 991, then n = 1009 and the condition k > n would be violated. Option 1000 and option 2000 are wrong because 999919 is not divisible evenly by 1000 or 2000, so they cannot be the exact number of rats killed by each cat while still giving a whole number of cats.

Common Pitfalls:
Students sometimes forget to use the condition that each cat kills more rats than there are cats, and they may simply pick any factor that appears reasonable. Another common error is to assume that the largest factor in the options must be the answer without checking both divisibility and the inequality k > n. Careful reading of the statement and systematic checking of the factor pairs helps avoid these mistakes.

Final Answer:
The correct number of rats killed by each cat is 1009.