Editing Divisibility rule (section)

==Beyond 30==
Divisibility properties of numbers can be determined in two ways, depending on the type of the divisor.

===Composite divisors===
A number is divisible by a given divisor if it is divisible by the highest power of each of its [[prime number|prime]] factors.  For example, to determine divisibility by 36, check divisibility by 4 and by 9.<ref name="product-of-coprimes"/>  Note that checking 3 and 12, or 2 and 18, would not be sufficient.  A [[table of prime factors]] may be useful.

A [[Composite number|composite]] divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor.  For instance, one cannot make a rule for 14 that involves multiplying the equation by 7.  This is not an issue for prime divisors because they have no smaller factors.

===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).