# 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:

 $\ds \paren {10 a + b} \paren {10 c + d}$ $=$ $\ds \paren {10 b + a} \paren {10 d + c}$ $\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 divisor count is $3$ or more, and find all those which have the product of single-digit integers in $2$ ways, as follows:

 $\ds \map {\sigma_0} 4$ $=$ $\ds 3$ $\sigma_0$ of $4$ $\ds \leadsto \ \$ $\ds 4$ $=$ $\ds 1 \times 4$ $\ds$ $=$ $\ds 2 \times 2$ $\ds \leadsto \ \$ $\ds 12 \times 42$ $=$ $\ds 21 \times 24$ $\ds$ $=$ $\ds 504$

 $\ds \map {\sigma_0} 6$ $=$ $\ds 4$ $\sigma_0$ of $6$ $\ds \leadsto \ \$ $\ds 6$ $=$ $\ds 1 \times 6$ $\ds$ $=$ $\ds 2 \times 3$ $\ds \leadsto \ \$ $\ds 12 \times 63$ $=$ $\ds 21 \times 36$ $\ds$ $=$ $\ds 756$ $\ds 13 \times 62$ $=$ $\ds 31 \times 26$ $\ds$ $=$ $\ds 806$

 $\ds \map {\sigma_0} 8$ $=$ $\ds 4$ $\sigma_0$ of $8$ $\ds \leadsto \ \$ $\ds 8$ $=$ $\ds 1 \times 8$ $\ds$ $=$ $\ds 2 \times 4$ $\ds \leadsto \ \$ $\ds 12 \times 84$ $=$ $\ds 21 \times 48$ $\ds$ $=$ $\ds 1008$ $\ds 14 \times 82$ $=$ $\ds 41 \times 28$ $\ds$ $=$ $\ds 1148$

 $\ds \map {\sigma_0} 9$ $=$ $\ds 3$ $\sigma_0$ of $9$ $\ds \leadsto \ \$ $\ds 9$ $=$ $\ds 1 \times 9$ $\ds$ $=$ $\ds 3 \times 3$ $\ds \leadsto \ \$ $\ds 13 \times 93$ $=$ $\ds 31 \times 39$ $\ds$ $=$ $\ds 1209$

 $\ds \map {\sigma_0} {10}$ $=$ $\ds 4$ $\sigma_0$ of $10$ $\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 $\map {\sigma_0} n \le 4$ need not be investigated, as one of the pairs of factors will be greater than $9$.

 $\ds \map {\sigma_0} {12}$ $=$ $\ds 6$ $\sigma_0$ of $12$ $\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 \ \$ $\ds 23 \times 64$ $=$ $\ds 32 \times 46$ $\ds$ $=$ $\ds 1472$ $\ds 24 \times 63$ $=$ $\ds 42 \times 36$ $\ds$ $=$ $\ds 1512$

 $\ds \map {\sigma_0} {16}$ $=$ $\ds 5$ $\sigma_0$ of $16$ $\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 \ \$ $\ds 24 \times 84$ $=$ $\ds 42 \times 48$ $\ds$ $=$ $\ds 2016$

 $\ds \map {\sigma_0} {18}$ $=$ $\ds 6$ $\sigma_0$ of $18$ $\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 \ \$ $\ds 23 \times 96$ $=$ $\ds 32 \times 69$ $\ds$ $=$ $\ds 2208$ $\ds 26 \times 93$ $=$ $\ds 62 \times 39$ $\ds$ $=$ $\ds 2418$

 $\ds \map {\sigma_0} {20}$ $=$ $\ds 6$ $\sigma_0$ of $20$ $\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 $\ds$ $=$ $\ds 4 \times 5$

 $\ds \map {\sigma_0} {24}$ $=$ $\ds 8$ $\sigma_0$ of $24$ $\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 \ \$ $\ds 34 \times 86$ $=$ $\ds 43 \times 68$ $\ds$ $=$ $\ds 2924$ $\ds 36 \times 84$ $=$ $\ds 63 \times 48$ $\ds$ $=$ $\ds 3024$

 $\ds \map {\sigma_0} {36}$ $=$ $\ds 9$ $\sigma_0$ of $36$ $\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 \ \$ $\ds 46 \times 96$ $=$ $\ds 64 \times 69$ $\ds$ $=$ $\ds 4416$