Scale Modularity

We define a scale modularity analogy to the concept of $\bmod$. This is modular in a product sense rather than an addition sense. Some properties are investigated. DEFINITION We have modular arithmetic and the mod operation can be explained as: $a \bmod b$ means take a, subtract b as many times as possible while keeping the difference greater than 0, and then return the result. We can generate a new statement: a scm b means take a, divide by b as many times as possible while keeping the quotient greater than 1, and then return the result. The additive inverse operation ‘subtract’ was changed to the multiplicative inverse ‘divide’ and the additive identity 0 was changed to the multiplicative identity 1. We can consider the properties of this operation: CONJECTURES Here codes like A123456 refer to sequences in the online encyclopedia of integer sequences (OEIS). Some conjectures on this operation are: $$ ab \; \; c = (a \; \; c)(b \; \; c)\; \; c $$ $$ a\cdots z \; \; c = (a \; \; c)\cdots(z \; \; c)\; \; c $$ $$ n \; \; 2 = {A082908(n-1)} $$ $$ p_k \; \; k \sim \log k $$ $$ 2^k \;\; 2 = 1 $$ $$ 3 \cdot 2^k \;\; 2 = {2} $$ $$ 5 \cdot 2^k \;\; 2 = {4} $$ $$ 2^k-1 \;\; 2 = 2 $$ $$ \; 2) 2^{\lceil \log_2 p_k\rceil}}{2p_k} = 1, k>1 $$