Finding the Smallest 4-Digit Number Divisible by Specific Numbers

When seeking the smallest 4-digit positive number that is divisible by the set of numbers {2, 3, 4, 5, 6, 8, 9, 10}, one must determine the least common multiple (LCM) of these numbers. Understanding the LCM helps in finding such a number. This article will guide you through the process and explain the mathematical concepts involved.

Prime Factorization and LCM Calculation

The first step in finding the LCM is to perform the prime factorization of each number in the set. Here are the prime factorizations:

2 21 3 31 4 22 5 51 6 21 × 31 8 23 9 32 10 21 × 51

Next, we determine the highest power of each prime factor across all the numbers:

The highest power of 2 is 23 from 8. The highest power of 3 is 32 from 9. The highest power of 5 is 51 from 5.

The LCM is then calculated by multiplying these highest powers together:

LCM 23 × 32 × 51 8 × 9 × 5 360

Determining the Smallest 4-Digit Number Divisible by 360

With the LCM known, the next step is to find the smallest 4-digit number that is divisible by 360. The smallest 4-digit number is 1000. To find this number, we divide 1000 by 360 and round up to the nearest whole number:

1000 ÷ 360 ≈ 2.7778

Rounding up gives us 3.

Multiplying this by 360 gives:

3 × 360 1080

Therefore, the smallest 4-digit positive number that is divisible by 2, 3, 4, 5, 6, 8, 9, and 10 is 1080.

Verification and Additional Insights

It's important to note that since the question asks for the smallest 4-digit number, the answer is unique and should be 1080. Any other answer would be incorrect.

For a brute force solution in the J programming language:

```j {.n~./02 3 4 5 6 8 9 10/n.1000 to 9999 1080 ```

This code snippet confirms our result by finding the smallest number in the range 1000 to 9999 that is divisible by all the given numbers.

Another interesting point is that 5 x 8 x 9 360, showing that the LCM can be derived in multiple ways.

Additionally, considering fractions and decimals, the answer would be 1 (if we are not constrained to whole numbers).

Conclusion

Understanding how to find the LCM and apply it to solve problems related to divisibility is a crucial skill in mathematics. Whether you're working on an academic problem or optimizing an SEO strategy for a website, familiarity with these concepts can be beneficial.

Key keywords for this article include least common multiple (LCM), 4-digit number, and divisible.