# Products of 2-Digit Pairs which Reversed reveal Same Product

## Theorem

The following positive integers can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:

- $504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2924, 3024, 4416$

## Proof

Let $n \in \Z_{>0}$ such that:

- $n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$

where $\sqbrk {a b}$ denotes the two-digit positive integer:

- $10 a + b$ for $0 \le a, b \le 9$

from the Basis Representation Theorem.

We have:

\(\ds \paren {10 a + b} \paren {10 c + d}\) | \(=\) | \(\ds \paren {10 b + a} \paren {10 d + c}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 100 a c + 10 \paren {a d + b c} + b d\) | \(=\) | \(\ds 100 b d + 10 \paren {b c + a d} + a c\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 99 a c\) | \(=\) | \(\ds 99 b d\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds a c\) | \(=\) | \(\ds b d\) |

Thus the problem boils down to finding all the sets of one-digit integers $\set {a, b, c, d}$ such that $a c = b d$, and so that:

- $n = \sqbrk {a b} \times \sqbrk {c d} = \sqbrk {b a} \times \sqbrk {d c}$

and also:

- $n = \sqbrk {a d} \times \sqbrk {b c} = \sqbrk {d a} \times \sqbrk {c b}$

Thus we investigate all integers whose $\tau$ value is $3$ or more, and find all those which have the product of single-digit integers in $2$ ways, as follows:

\(\ds \map \tau 4\) | \(=\) | \(\ds 3\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 4\) | \(=\) | \(\ds 1 \times 4\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 2\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 12 \times 42\) | \(=\) | \(\ds 21 \times 24\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 504\) |

\(\ds \map \tau 6\) | \(=\) | \(\ds 4\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 6\) | \(=\) | \(\ds 1 \times 6\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 3\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 12 \times 63\) | \(=\) | \(\ds 21 \times 36\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 756\) | ||||||||||||

\(\ds 13 \times 62\) | \(=\) | \(\ds 31 \times 26\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 806\) |

\(\ds \map \tau 8\) | \(=\) | \(\ds 4\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 8\) | \(=\) | \(\ds 1 \times 8\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 4\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 12 \times 84\) | \(=\) | \(\ds 21 \times 48\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 1008\) | ||||||||||||

\(\ds 14 \times 82\) | \(=\) | \(\ds 41 \times 28\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1148\) |

\(\ds \map \tau 9\) | \(=\) | \(\ds 3\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 9\) | \(=\) | \(\ds 1 \times 9\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 3 \times 3\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 13 \times 93\) | \(=\) | \(\ds 31 \times 39\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 1209\) |

\(\ds \map \tau {10}\) | \(=\) | \(\ds 4\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 10\) | \(=\) | \(\ds 1 \times 10\) | and so does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 5\) |

Further integers $n$ such that $\map \tau n \le 4$ need not be investigated, as one of the pairs of factors will be greater than $9$.

\(\ds \map \tau {12}\) | \(=\) | \(\ds 6\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 12\) | \(=\) | \(\ds 1 \times 12\) | which does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 6\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 3 \times 4\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 23 \times 64\) | \(=\) | \(\ds 32 \times 46\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 1472\) | ||||||||||||

\(\ds 24 \times 63\) | \(=\) | \(\ds 42 \times 36\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 1512\) |

\(\ds \map \tau {16}\) | \(=\) | \(\ds 5\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 16\) | \(=\) | \(\ds 1 \times 16\) | which does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 8\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 4 \times 4\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 24 \times 84\) | \(=\) | \(\ds 42 \times 48\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 2016\) |

\(\ds \map \tau {18}\) | \(=\) | \(\ds 6\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 18\) | \(=\) | \(\ds 1 \times 18\) | which does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 9\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 3 \times 6\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 23 \times 96\) | \(=\) | \(\ds 32 \times 69\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 2208\) | ||||||||||||

\(\ds 26 \times 93\) | \(=\) | \(\ds 62 \times 39\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2418\) |

\(\ds \map \tau {20}\) | \(=\) | \(\ds 6\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 20\) | \(=\) | \(\ds 1 \times 20\) | which does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 10\) | which does not lead to a solution | |||||||||||

\(\ds \) | \(=\) | \(\ds 4 \times 5\) |

\(\ds \map \tau {24}\) | \(=\) | \(\ds 8\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 24\) | \(=\) | \(\ds 1 \times 24\) | which does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 12\) | which does not lead to a solution | |||||||||||

\(\ds \) | \(=\) | \(\ds 3 \times 8\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 4 \times 6\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 34 \times 86\) | \(=\) | \(\ds 43 \times 68\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 2924\) | ||||||||||||

\(\ds 36 \times 84\) | \(=\) | \(\ds 63 \times 48\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 3024\) |

\(\ds \map \tau {36}\) | \(=\) | \(\ds 9\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 36\) | \(=\) | \(\ds 1 \times 36\) | which does not lead to a solution | ||||||||||

\(\ds \) | \(=\) | \(\ds 2 \times 18\) | which does not lead to a solution | |||||||||||

\(\ds \) | \(=\) | \(\ds 3 \times 12\) | which does not lead to a solution | |||||||||||

\(\ds \) | \(=\) | \(\ds 4 \times 9\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 6 \times 6\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 46 \times 96\) | \(=\) | \(\ds 64 \times 69\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 4416\) |

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $504$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $504$