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

## 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:

 $\displaystyle 144 \, 648$ $=$ $\displaystyle 861 \times 168$ $\displaystyle$ $=$ $\displaystyle 492 \times 294$

## 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.