Products of 2-Digit Pairs which Reversed reveal Same Product

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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:

\(\displaystyle \paren {10 a + b} \paren {10 c + d}\) \(=\) \(\displaystyle \paren {10 b + a} \paren {10 d + c}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 100 a c + 10 \paren {a d + b c} + b d\) \(=\) \(\displaystyle 100 b d + 10 \paren {b c + a d} + a c\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 99 a c\) \(=\) \(\displaystyle 99 b d\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle a c\) \(=\) \(\displaystyle b d\)


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:

\(\displaystyle \map \tau 4\) \(=\) \(\displaystyle 3\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 4\) \(=\) \(\displaystyle 1 \times 4\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 12 \times 42\) \(=\) \(\displaystyle 21 \times 24\)
\(\displaystyle \) \(=\) \(\displaystyle 504\)


\(\displaystyle \map \tau 6\) \(=\) \(\displaystyle 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 6\) \(=\) \(\displaystyle 1 \times 6\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 3\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 12 \times 63\) \(=\) \(\displaystyle 21 \times 36\)
\(\displaystyle \) \(=\) \(\displaystyle 756\)
\(\displaystyle 13 \times 62\) \(=\) \(\displaystyle 31 \times 26\)
\(\displaystyle \) \(=\) \(\displaystyle 806\)


\(\displaystyle \map \tau 8\) \(=\) \(\displaystyle 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 8\) \(=\) \(\displaystyle 1 \times 8\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 12 \times 84\) \(=\) \(\displaystyle 21 \times 48\)
\(\displaystyle \) \(=\) \(\displaystyle 1008\)
\(\displaystyle 14 \times 82\) \(=\) \(\displaystyle 41 \times 28\)
\(\displaystyle \) \(=\) \(\displaystyle 1148\)


\(\displaystyle \map \tau 9\) \(=\) \(\displaystyle 3\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 9\) \(=\) \(\displaystyle 1 \times 9\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 3\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 13 \times 93\) \(=\) \(\displaystyle 31 \times 39\)
\(\displaystyle \) \(=\) \(\displaystyle 1209\)


\(\displaystyle \map \tau {10}\) \(=\) \(\displaystyle 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 10\) \(=\) \(\displaystyle 1 \times 10\) and so does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 5\)


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 $9$.


\(\displaystyle \map \tau {12}\) \(=\) \(\displaystyle 6\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 12\) \(=\) \(\displaystyle 1 \times 12\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 6\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 23 \times 64\) \(=\) \(\displaystyle 32 \times 46\)
\(\displaystyle \) \(=\) \(\displaystyle 1472\)
\(\displaystyle 24 \times 63\) \(=\) \(\displaystyle 42 \times 36\)
\(\displaystyle \) \(=\) \(\displaystyle 1512\)


\(\displaystyle \map \tau {16}\) \(=\) \(\displaystyle 5\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 16\) \(=\) \(\displaystyle 1 \times 16\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 8\)
\(\displaystyle \) \(=\) \(\displaystyle 4 \times 4\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 24 \times 84\) \(=\) \(\displaystyle 42 \times 48\)
\(\displaystyle \) \(=\) \(\displaystyle 2016\)


\(\displaystyle \map \tau {18}\) \(=\) \(\displaystyle 6\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 18\) \(=\) \(\displaystyle 1 \times 18\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 9\)
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 6\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 23 \times 96\) \(=\) \(\displaystyle 32 \times 69\)
\(\displaystyle \) \(=\) \(\displaystyle 2208\)
\(\displaystyle 26 \times 93\) \(=\) \(\displaystyle 62 \times 39\)
\(\displaystyle \) \(=\) \(\displaystyle 2418\)


\(\displaystyle \map \tau {20}\) \(=\) \(\displaystyle 6\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 20\) \(=\) \(\displaystyle 1 \times 20\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 10\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 4 \times 5\)


\(\displaystyle \map \tau {24}\) \(=\) \(\displaystyle 8\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 24\) \(=\) \(\displaystyle 1 \times 24\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 12\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 8\)
\(\displaystyle \) \(=\) \(\displaystyle 4 \times 6\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 34 \times 86\) \(=\) \(\displaystyle 43 \times 68\)
\(\displaystyle \) \(=\) \(\displaystyle 2924\)
\(\displaystyle 36 \times 84\) \(=\) \(\displaystyle 63 \times 48\)
\(\displaystyle \) \(=\) \(\displaystyle 3024\)


\(\displaystyle \map \tau {36}\) \(=\) \(\displaystyle 9\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 36\) \(=\) \(\displaystyle 1 \times 36\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 2 \times 18\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 3 \times 12\) which does not lead to a solution
\(\displaystyle \) \(=\) \(\displaystyle 4 \times 9\)
\(\displaystyle \) \(=\) \(\displaystyle 6 \times 6\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 46 \times 96\) \(=\) \(\displaystyle 64 \times 69\)
\(\displaystyle \) \(=\) \(\displaystyle 4416\)




Sources