# Integers Expressible as Product of Number and Reversal in 2 Different Ways

Jump to navigation
Jump to search

## Contents

## Theorem

The number $2520$ is the smallest which can be expressed as the product of a number and its reversal in two different ways:

\(\displaystyle 2520\) | \(=\) | \(\displaystyle 210 \times 012\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 120 \times 021\) |

The number $63 \, 504$ is the smallest which is not a multiple of $10$:

\(\displaystyle 63 \, 504\) | \(=\) | \(\displaystyle 441 \times 144\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 252 \times 252\) |

The number $7 \, 683 \, 984$ is another such which includes a palindrome and is therefore square:

\(\displaystyle 7 \, 683 \, 984\) | \(=\) | \(\displaystyle 2772 \times 2772\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1584 \times 4851\) |

The number $144 \, 648$ does not include a palindrome:

\(\displaystyle 144 \, 648\) | \(=\) | \(\displaystyle 861 \times 168\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 492 \times 294\) |

## Proof

## Historical Note

In his $1997$ work *Curious and Interesting Numbers, 2nd ed.*, David Wells attributes this result to S.S. Gupta, but provides no details.

It remains to establish the source.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2520$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $63,504$