Exploring a Mathematical Puzzle: Finding a Number Divisible by 1 Through 10 with Specific Remainders

Mathematics is filled with intriguing problems that challenge and enlighten its practitioners. One such problem asks: what is a number that when divided by 10 leaves a remainder 9, when divided by 9 leaves a remainder 8, and so on, down to when divided by 2 leaves a remainder 1? In this article, we delve into the solution to this fascinating problem, discussing the mathematical techniques and logical steps involved, and we'll also demonstrate how this solution is consistent across different computational methods.

Understanding the Problem

The problem can be stated as finding a number (x) such that:

(x equiv -1 mod 10) (x equiv -1 mod 9) (x equiv -1 mod 8) and so on, until (x equiv -1 mod 2)

This can be generalized to say that (x) must be of the form (x equiv -1 mod n), where (n) ranges from 10 down to 2. In other words, (x 1) must be a common multiple of all these numbers.

Step-by-Step Solution

To solve this, we need to find the least common multiple (LCM) of the numbers 10 through 2. Let's break this down.

Prime Factorization and LCM Calculation

Prime factorizations are as follows:

(10 2 times 5) (9 3^2) (8 2^3) (7 7) (6 2 times 3) (5 5) (4 2^2) (3 3) (2 2)

The highest powers of each prime are:

(2^3) (3^2) (5^1) (7^1)

Thus, the LCM is calculated as:

[ text{LCM} 2^3 times 3^2 times 5 times 7 8 times 9 times 5 times 7 2520 ]

Final Calculation

Since (x 1 2520), we have:

[ x 2520 - 1 2519 ]

Conclusion

The final answer to the given problem is the number 2519. This number satisfies the conditions specified: when divided by 10, 9, 8, ..., 2, it leaves remainders 9, 8, 7, ..., 1, respectively.

Verification Using Different Computational Methods

Another approach to verify the solution is by using the modulo arithmetic representation directly:

[ x -1 mod 9875 -1 mod 2520 ]

This verifies that 2519 is indeed the smallest number that satisfies all the given conditions.

A Deeper Insight into Number Theory

This problem is a prime example of the application of number theory and the concept of least common multiple in solving complex mathematical problems. The solution to such a problem can be extended to various other scenarios where similar patterns in remainders need to be addressed.

Understanding and solving such puzzles not only enhances one's problem-solving skills but also deepens the appreciation of number theory and its practical applications.

Key Takeaways

The problem involves finding a number that leaves specific remainders when divided by consecutive integers from 10 down to 2. The solution utilizes the concept of least common multiple to determine the smallest number that satisfies the conditions. The final answer is 2519, which is derived from the LCM of the numbers 10 through 2.

Keywords: number theory, mathematical problem, least common multiple