A recent meme showed the following "algebra":
$$\frac{777}{4773} = \frac{7(77)}{4*(77)*3}=\frac{7}{43}$$
which is, amazingly, true!
Clearly, there are infinitely many natural number pairs that satisfy this property (by adding on a zero to numerator and denominator, hence multiplying by 10), but we ignore those solutions as trivial.
We define $\text{value}(a_M,a_{M-1},...,a_1,a_0) = \sum_{m=0}^M a_m 10^m$
let $A, B \in \mathbb{N}$ be given. We then say that $A = \text{value}(a_M,a_{M-1},...,a_1,a_0)$ and $B= \text{value}(b_N,b_{N-1},...,b_1,b_0)$ are dodgy-reducible if
$$\frac{\text{value}(a_M,a_{M-1},...,a_1,a_0)}{\text{value}(b_N,b_{N-1},...,b_1,b_0)}=\frac{\text{value}(a_{k_i},a_{{k_{i-1}}},...,a_{k_1},a_{k_0})}{\text{value}(b_{l_j},b_{{l_{j-1}}},...,b_{l_1},b_{l_0})}$$
where the $k_i$s and the $l_i$s are any selection of the numbers $\{0,1,...,M\}$ and $\{0,1,...,N\}$ respectively.
The first handful are:
64 16
65 26
95 19
98 49
121 22
132 33
136 34
143 44
154 55
165 66
176 77
184 138
185 148
187 88
192 96
194 97
195 39
196 49
196 98
198 99
I have tried (and failed) to make any analytic headway, but my friend wrote a program to naïvely iterate through all integer pairs below a given threshold, yielding the following distribution:
0 - 300 (excluding trivial solutions) 0-300 0-~4800 (excluding trivial solutions) 0-4800
The question: Is this a special case of a known sequence in number theory, and if not, what can be determined about its distribution?
