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

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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