Exploring the Nine-Digit Number Divisible by 1 to 9

In mathematics, finding a nine-digit number that is divisible by all the digits from 1 to 9 is an intriguing problem. This article delves into the solution and provides a comprehensive understanding of the concept and its computation.

Let us begin with the known multiplication facts. The product of the digits 1 to 9 is:

Product of Digits 1 to 9

First, we calculate:

9 × 8 × 7 × 5

The result is 2,520. This number forms the base of our solution.

Divisibility and Range of Nine-Digit Numbers

Given that 2,520 is the product of the digits 1 to 9, we aim to find the smallest nine-digit number that is a multiple of 2,520. To do this, we must consider the range of nine-digit numbers, which starts from 100,000,000 and goes up to 999,999,999.

Calculation of the Nine-Digit Multiples

The smallest nine-digit multiple of 2,520 is:

[ 2,520 times leftlceil frac{10^8}{2,520} rightrceil ]

Calculating the ceiling function:

[ 10^8 100,000,000 ] [ frac{100,000,000}{2,520} approx 39,683.25 ] [ leftlceil 39,683.25 rightrceil 39,684 ]

Thus, the smallest nine-digit multiple is:

[ 2,520 times 39,684 99,999,936 ]

The largest nine-digit multiple of 2,520 is:

[ 2,520 times leftlfloor frac{10^9 - 1}{2,520} rightrfloor ]

Calculating the floor function:

[ 10^9 - 1 999,999,999 ] [ frac{999,999,999}{2,520} approx 396,825.39 ] [ leftlfloor 396,825.39 rightrfloor 396,825 ]

Thus, the largest nine-digit multiple is:

[ 2,520 times 396,825 999,999,000 ]

Therefore, the nine-digit numbers divisible by the digits 1 to 9 form a range from 99,999,936 to 999,999,000, inclusive.

Counting the Numbers in the Range

To find the total count of such nine-digit numbers, we subtract the smallest number from the largest number and add 1 to include both endpoints:

[ 396,825 - 39,684 1 357,142 ]

This means there are 357,142 such nine-digit numbers. It is important to note that this count includes all positive integers within the specified range.

Conclusion

In conclusion, the nine-digit number divisible by 1 to 9 is 999,999,000, and there are 357,142 such numbers. This exploration not only offers a deep insight into the divisibility properties of numbers but also demonstrates the application of mathematical concepts in solving complex problems.