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: {{begin-eqn}} {{eqn | l = \paren {10 a + b} \paren {10 c + d}       | r = \paren {10 b + a} \paren {10 d + c}       | c =  }} {{eqn | ll= \leadsto       | l = 100 a c + 10 \paren {a d + b c} + b d       | r = 100 b d + 10 \paren {b c + a d} + a c       | c =  }} {{eqn | ll= \leadsto       | l = 99 a c        | r = 99 b d       | c =  }} {{eqn | ll= \leadsto       | l = a c        | r = b d       | c =  }} {{end-eqn}} 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: {{begin-eqn}} {{eqn | l = \map \tau 4       | r = 3       | c =  }} {{eqn | ll= \leadsto       | l = 4       | r = 1 \times 4       | c =  }} {{eqn | r = 2 \times 2       | c =  }} {{eqn | ll= \leadsto       | l = 12 \times 42       | r = 21 \times 24       | c =  }} {{eqn | r = 504       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau 6       | r = 4       | c =  }} {{eqn | ll= \leadsto       | l = 6       | r = 1 \times 6       | c =  }} {{eqn | r = 2 \times 3       | c =  }} {{eqn | ll= \leadsto       | l = 12 \times 63       | r = 21 \times 36       | c =  }} {{eqn | r = 756       | c =  }} {{eqn | l = 13 \times 62       | r = 31 \times 26       | c =  }} {{eqn | r = 806       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau 8       | r = 4       | c =  }} {{eqn | ll= \leadsto       | l = 8       | r = 1 \times 8       | c =  }} {{eqn | r = 2 \times 4       | c =  }} {{eqn | ll= \leadsto       | l = 12 \times 84       | r = 21 \times 48       | c =  }} {{eqn | r = 1008       | c =  }} {{eqn | l = 14 \times 82       | r = 41 \times 28       | c =  }} {{eqn | r = 1148       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau 9       | r = 3       | c =  }} {{eqn | ll= \leadsto       | l = 9       | r = 1 \times 9       | c =  }} {{eqn | r = 3 \times 3       | c =  }} {{eqn | ll= \leadsto       | l = 13 \times 93       | r = 31 \times 39       | c =  }} {{eqn | r = 1209       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau {10}       | r = 4       | c =  }} {{eqn | ll= \leadsto       | l = 10       | r = 1 \times 10       | c = and so does not lead to a solution }} {{eqn | r = 2 \times 5       | c =  }} {{end-eqn}} 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 $9$. {{begin-eqn}} {{eqn | l = \map \tau {12}       | r = 6       | c =  }} {{eqn | ll= \leadsto       | l = 12       | r = 1 \times 12       | c = which does not lead to a solution }} {{eqn | r = 2 \times 6       | c =  }} {{eqn | r = 3 \times 4       | c =  }} {{eqn | ll= \leadsto       | l = 23 \times 64       | r = 32 \times 46       | c =  }} {{eqn | r = 1472       | c =  }} {{eqn | l = 24 \times 63       | r = 42 \times 36       | c =  }} {{eqn | r = 1512       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau {16}       | r = 5       | c =  }} {{eqn | ll= \leadsto       | l = 16       | r = 1 \times 16       | c = which does not lead to a solution }} {{eqn | r = 2 \times 8       | c =  }} {{eqn | r = 4 \times 4       | c =  }} {{eqn | ll= \leadsto       | l = 24 \times 84       | r = 42 \times 48       | c =  }} {{eqn | r = 2016       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau {18}       | r = 6       | c =  }} {{eqn | ll= \leadsto       | l = 18       | r = 1 \times 18       | c = which does not lead to a solution }} {{eqn | r = 2 \times 9       | c =  }} {{eqn | r = 3 \times 6       | c =  }} {{eqn | ll= \leadsto       | l = 23 \times 96       | r = 32 \times 69       | c =  }} {{eqn | r = 2208       | c =  }} {{eqn | l = 26 \times 93       | r = 62 \times 39       | c =  }} {{eqn | r = 2418       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau {20}       | r = 6       | c =  }} {{eqn | ll= \leadsto       | l = 20       | r = 1 \times 20       | c = which does not lead to a solution }} {{eqn | r = 2 \times 10       | c = which does not lead to a solution }} {{eqn | r = 4 \times 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau {24}       | r = 8       | c =  }} {{eqn | ll= \leadsto       | l = 24       | r = 1 \times 24       | c = which does not lead to a solution }} {{eqn | r = 2 \times 12       | c = which does not lead to a solution }} {{eqn | r = 3 \times 8       | c =  }} {{eqn | r = 4 \times 6       | c =  }} {{eqn | ll= \leadsto       | l = 34 \times 86       | r = 43 \times 68       | c =  }} {{eqn | r = 2924       | c =  }} {{eqn | l = 36 \times 84       | r = 63 \times 48       | c =  }} {{eqn | r = 3024       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map \tau {36}       | r = 9       | c =  }} {{eqn | ll= \leadsto       | l = 36       | r = 1 \times 36       | c = which does not lead to a solution }} {{eqn | r = 2 \times 18       | c = which does not lead to a solution }} {{eqn | r = 3 \times 12       | c = which does not lead to a solution }} {{eqn | r = 4 \times 9       | c =  }} {{eqn | r = 6 \times 6       | c =  }} {{eqn | ll= \leadsto       | l = 46 \times 96       | r = 64 \times 69       | c =  }} {{eqn | r = 4416       | c =  }} {{end-eqn}} {{ProofWanted|The challenge remains to prove, without going through all cases exhaustively, that there are no further pairs.}} {{Improve|Why not write out the upper triangular part of the multiplication table for $1 \le n \le 9$, $n \ne 5, 7$, and find the common numbers in the table? </br> $5, 7$ can be eliminated via a simple argument $p x \ge 10$, and there are only $28$ numbers left to compare}}