Cab Numbers

The concept of Cab Numbers, introduced by Henry E. Dudeney, involves finding two numbers that, when multiplied, yield a product containing all nine digits (1-9) without repetition. The highest known example is 8745231 multiplied by 96, resulting in 839542176. Dudeney's work also provides solutions for three, four, and five-digit Cab Numbers, with specific examples listed. Cab Numbers are defined as numbers made up of distinct digits (excluding zero) that can be expressed as the product of two factors containing the same digits. A computer program written in Fortran has computed solutions for six to nine-digit Cab Numbers, revealing properties such as the digital root, which must match between the factors and their product. The program identified 1625 solutions for nine-digit Cab Numbers and 12449 for ten-digit solutions, including those with zero in an internal position. Further investigations have explored the multiplication of more than two factors to produce numbers with the same digits, yielding additional solutions. The study of Cab Numbers highlights interesting mathematical properties and relationships among digits.

- Cab Numbers consist of distinct digits and can be expressed as the product of two numbers with the same digits.

- The highest known Cab Number example is 8745231 * 96 = 839542176.

- A Fortran program computed solutions for Cab Numbers with up to ten digits, identifying thousands of valid combinations.

- Digital roots play a crucial role in determining valid Cab Numbers.

- Further research includes exploring products of more than two factors that yield numbers with the same digits.