In the November 2003 Focus Paul N. Cohn briefly reprises some neat old
notes on mental divisibility tests (the references are dated 1861, 1919,
1938 and 1943) that may be of interest to math-funsters.
An example: to test divisibility by 7 subtract twice the last digit from
the rest, ie 10a+b --> a-2b, until the result is apparent by
inspection. For instance, 7 | 2072-->203-->14 (or --> -7, if 14 is still
too inscrutable).
He notes this idea generalizes to divisors having multiples of the form
10s+1 or 10s-1 (eg s=1 gives the traditional casting out 9s and 11s, 7
works because 3*7=21, 13 can be tested by adding 4b, etc).
Observe that in base 2 (instead of decimal) the test reduces to (part of)
the binary GCD algorithm.
A handy hack, possibly suggestive of further fun?
