CONGRUENCE 89

For example, if 25 is divided by 4,
for the quotient 4 we have the remamder 9

‘e [ 5 ol ‘e fl ‘il 5

(s ‘é ‘e 6 fl X [ ‘é 1

6’ 6l e 7 “l (X3 6 ‘l -t 3

il “l ‘¢ 8 6l [ ‘e ‘¢ A 7
Hence 25 =9=5=1= — 3= — 7 (mod 4)

Again, when 32 is divided by 4,

with the quot1ent 7 we get the remamder 4
‘il fé 8 ‘il é ‘¢ [ 0
‘e [ ol 9 X3 ‘é ‘ol (X3 DL 4:
Hence 32 =4 = 0 = — 4 (mod 4)

For the converse theorem, since @ = r (mod m), a — r > m,
whence ¢ — r = gm and r = @ — gm.

Therfor 7 is the remainder corresponding to the quotient ¢
when a is divided by m.

203. Th. If two integers leave the same remainder when
divided by the modulus, they are congruent; and conversely, if
two integers are congruent, they leave the same remainders when
divided by the modulus.

Suppose a and b leave the remainder 7.

Thena =rand b = r. § 202.

Hence a = b. §§ 185, 188.

For the converse, suppose b leaves the remainder r when
divided by m.

Then b = r (mod m). § 202.

Hence, if @ = b (mod m), @ = r (mod m).

Then 7 is a remainder obtaind by dividing @ by m.  § 202.

204. Th. (ea=b) =(@+c=b+0)
For(a — b3 m) =[(a+c)— (b+c)>m]

205. Th. (a=0b) =(c+a=c+D)
206 Th. a=b~c=d>Da+c=b+4d