We are tasked with finding the largest four-digit number that satisfies the following conditions:

• When divided by 2, the remainder is 1.
• When divided by 3, the remainder is 2.
• When divided by 4, the remainder is 3.
• When divided by 5, the remainder is 4.
• When divided by 6, the remainder is 5.
• When divided by 7, the remainder is 6.
• When divided by 8, the remainder is 7.
• When divided by 9, the remainder is 8.
• When divided by 10, the remainder is 9.

This can be summarized as finding the largest 4-digit integer $N$ such that:

$N \equiv -1 \ (\text{mod} \ k) \ \text{for} \ k = 2, 3, 4, \ldots, 10$

This is equivalent to finding a number that satisfies:

$N + 1 \text{ is divisible by } \ 2, 3, 4, 5, 6, 7, 8, 9, \text{and} \ 10.$

To solve this, we need to find the least common multiple (LCM) of the numbers 2 through 10. This will help us find $N + 1$, and from there, we can subtract 1 to get $N$.

Let me compute the LCM of 2 through 10 and find the largest four-digit number that satisfies this condition.The least common multiple (LCM) of the numbers from 2 to 10 is 2520. The largest four-digit number that satisfies the given conditions is:

$N = 7559$

This means that when 7559 is divided by 2, 3, 4, 5, 6, 7, 8, 9, or 10, it leaves remainders of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively.

Would you like a more detailed explanation of the steps?

Here are 5 related questions to further explore this topic:

1. How can you verify that 7559 satisfies all the given conditions?
2. What is the significance of finding the LCM in problems like this?
3. How would the approach change if we needed a 5-digit number instead?
4. Can we use this method to find smaller numbers with similar properties?
5. Why does subtracting 1 from the LCM-based result give us the correct solution?

Tip: The least common multiple (LCM) is useful for solving problems where numbers must be divisible by several integers simultaneously.