Editing Divisibility rule (section)

===Prime divisors===
The goal is to find an inverse to 10 [[modular arithmetic|modulo]] the prime under consideration (does not work for 2 or 5) and use that as a multiplier to make the divisibility of the original number by that prime depend on the divisibility of the new (usually smaller) number by the same prime.
Using 31 as an example, since 10 × (−3) = −30 = 1 mod 31, we get the rule for using ''y''&nbsp;−&nbsp;3''x'' in the table below. Likewise, since 10 × (28) = 280 = 1 mod 31 also, we obtain a complementary rule ''y''&nbsp;+&nbsp;28''x'' of the same kind - our choice of addition or subtraction being dictated by arithmetic convenience of the smaller value. In fact, this rule for prime divisors besides 2 and 5 is ''really'' a rule for divisibility by any integer relatively prime to 10 (including 33 and 39; see the table below).  This is why the last divisibility condition in the tables above and below for any number relatively prime to 10 has the same kind of form (add or subtract some multiple of the last digit from the rest of the number).