II

Con gruences

Every integer may be expressed as the multiple of a lower
integer plus a remainder. The number 7 can accordingly be
expressed as

L A e

qm-—-r
where 7 is the remainder when 7 is divided by m; gm being
a multiple of m. Where 7 is expressed as a multiple of m, and
m is a factor of n, the remainder will obviously be equal to
nought.

The number 7 will not, however, be the only number
which gives the remainder r when divided by m, and this is
the basic fact upon which the theory of congruences was
founded by the mathematician Gauss.

The numbers 40 and 64, for instance, give the same re-
mainder of 4 when divided by r2. This is another way of
saying that both numbers are equal to multiples of 12 added
to 4. They both, therefore, bear a certain common relation-
ship in their attitude to the number 12. In mathematical
language the latter number is called the ‘modulus’ in respect
1 of which the two other numbers have similar properties, and
‘ these two numbers are said to be ‘congruent’ to each other
for that modulus.

i Thus, if a=qm-r
and b=qm-+tr

i

e L L T R T Tt
P Pebite AR ORHG: 4 Eaae 1 Y

RREEE T T s s R R R

Hikini

"

then « is said to be congruent to & for the modulus m, and the
relationship is expressed as:

L T T AR
il FEESTTLY ST VE PR T O RMRPe RO Rve YL v o

a=b(mod m)

The quantities ¢ and b are termed residues of each other for
108

ORI

  

ST st MR i 0 105
i . oy b K