CONGRUENCE 91

Or a:c=b:c(modm:c)
For the converse theorem reverse the steps.

219. Th. If n is a natural number or zero and a = b, then
a® = b".

This follows, if #» > 1, by multiplying together n congru-
ences, each of which is a = b.

For example, a = b D a? = b ~ a® = b°

220. Th. If f(x) is a quantic and x = r, then flx) = f(r).
We have

n+1
flx) = 2 Gng-pi el § 104.
k=1
Since x = 7, g (k=1 = pn—@*-1) § 219.
Hence Qa0 P V= gy et B8 § 213.
n+1 n+1
Hence 3 Gngnx™ %D =3 g0t * Y § 207.
k=1 k=1 g
Or f(x) = f(r)

For example, if f(x) = x® + x* — 4,
ifx=0,f(x) =0 =—4
fx=1fx)=f1)=—2

221. Th. If f(x) is a quantic and r = s, then f(r) = f(s)
For example, if f(x) = x? — 3x — 2,
since 8 = 0 (mod 4), f(8) = f(0) (mod 4), or 38 = — 2 (mod 4)
since 34 = 4 (mod 6), f(34) = f(4) (mod 6), or 1052 = 2 (mod 6)

I

222. Th. Similarly, if f is a quantic operation with integral
coefficients upon any mumber of operands x, ¥, 2, **° and
flx, v, 2, =) represents the result of the operation, then,
if X5 Gy N = b, z=¢, **°, f(x7 Y, % "') Ef(a’v b, c, "')

For example,
if f(x,y) =22+ xy+rix=1l,andy =1,

then f(x, y) = f(1,1) = 3
iff(x,y,z) = (x ) y)(y o Z)(Z = x))x = 0,3' = 1,2 == 1!
then f(x, ¥, 2) = 2