Divisibility

Divisibility, in the Theory of Numbers, is that property of any number whereby it may be divided by another without remainder. To find the condition of divisibility of one number, $N$, by another, $D$. Let $N\; =\; b\_0\; +\; b\_1\; r\; +\; b\_2\; r^2\; +\; \backslash dots\; +\; b\_\{m-1\}\; r^\{m-1\}\; +\; b\_m\; r^m$, where are coefficients, and $r$ is the radix of the notational scale (see NOTATION). Introducing $D$ and $-D$ along with $r$, this may be written: $N\; =\; b\_0\; +\; b\_1\; (D\; +\; r\; -\; D)\; +\; b\_2\; (D\; +\; r\; -\; D)^2\; +\; \backslash dots\; +\; b\_m\; (D\; +\; r\; -\; D)^m$. Expanding the terms on the right-hand side of this equation, it will appear that $\backslash frac\{N\}\{D\}$ will be an integer if $b\_0\; +\; b\_1\; (r\; -\; D)\; +\; \backslash dots\; +\; b\_m\; (r\; -\; D)^m$ be divisible by $D$. For example, if $r\; =\; 10$ (i.e. if the number be given in the denary or ordinary scale), and $D\; =\; 9$, and therefore $r\; -\; D\; =\; 1$, any number will be divisible by 9 if the sum of its coefficients be so—i.e. if the sum of its digits be divisible by 9. Further rules found in this manner are fully given in Mackay's Arithmetical Exercises.