CONGRUENCES 111

must be a multiple of 10, plus 3; the other must be a multiple
of 10, plus 2. If these two residues are multiplied we would
have

(10, +3)(10,+2)
but (10, +3) would be an odd number, so that if 16=(10, +3)
(10,4-2), then 16 would be divisible by an odd number and
this is ridiculous.

The theory of congruences is greatly to the fore in general
number theory and has been called into use in many of the
problems involved in the discovery of tests for primality. As
an example, we have Wilson’s theorem that if p is prime then
(p—1)!+1=0(mod p); or conversely that if (p—1)!+1=0
(mod p) then p is prime.

As an example, we take the prime value p=r11.

Then (p—1)!+1=10!41=3628801
This number is divisible by 11, so

36288o1=0(mod 11)
and (p—1)!+1=0(mod 11)
Although this affords a test for primality, the very large
" numbers involved are so great as to render it unpractical.
Adaptations of other congruence theorems, mainly the result
of the work of the French mathematician Lucas, give a test
for primality but have the failing that they do not reveal the
actual factors of numbers shown to be composite.

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Wuun,;:w.u‘-mu%”akmm%aumwl