174 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

439. If @, b, and m are integers and m —= 0, then, since
ab — cd = (e + f)m,
ab —cd>m

Hence ab = cd (mod m)

Th. That is, if the base is a non-zero integer, the product of
two integers is congruent to the product of their complements with
respect to the base as a modulus.

440. If a, b, and # are integers, so are e and f.

Hence, since ab — ef = [n + (e + f) In,
if n—=0, ab = ef (mod n)

Th. That is, if half the base is a non-zero integer, the product
of two integers is congruent to the product of their excesses over
that half base with respect to the half base as a modulus.

441. Th.

a + b= m according as ¢ + d=m and acc. as e + f = 0
‘¢ ‘“ -_— ‘“ ‘ ‘6 —
a+b=2m c+d=0 e+ f=m

Forc+d=2m — (a+b)ande+f=(a+b) —m

442. Th. Hence
m<a+bdb<2m)=0<c+d<m)=0<e+f<m)
Of course here, since m < 2m, m must be positiv.

443. Th.
mM=a+b<2m)=0<c+d=m)=0=Ze+f<m)
444. Th.
n§a<mflnib<m:)(0<c+d§m)fi'(0§ e+ f<m)

For,ifa = n = b,thena + b =m;andifa > nand b = n,
ora=nand b >n,ora >nand b > n, thena 4+ b > m.