104 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Hence 2(24) + (— 1)(45) = 3 = 24 (X 45.

246. Th. If a and b are two integers prime to each other, two
integers 1 and m can be found, such that

la + mb = 1.
For example, taking ¢ = 9 and b = 7, we find
(—3)X9+4X7=1

247. Th. If a, b, and c are integers and a pair of integers
l, m can be found such that la + mb = c, an infinit number of
such pairs can be found.

First, if @ and b are both zero, since it is given that
la + mb = ¢, evidently ¢ must also be zero.

Hence, if I’, m’ are any pair of integers whatever,
l'a + m'b will equal ¢, which proves the theorem for this case.

Now suppose that e and b are not both zero.

Then either a—= 0 or b —= 0.

Suppose that ¢ —= 0.

Then, if z and 2’ are two different integers, that is, if

/

g2—= 3
az —= ag'
Then m 4+ az—= m + a2’

Then the pair of integers I — bz, m + az will be different
from the pair I — b2’, m + a2’.

But (I — bz)a + (m + az)b = la + mb

Then, since la + mb = ¢, (I — bz)a + (m + a2z)b = ¢

Similarly (I—0bz)a+ (m+ ad')b=c

Thus, if we have one pair (I, m) for which la + mb = ¢, we
can find, by choosing various values for 2, as many different
pairs (I — bz, m + az), (I — bd’, m + a2’), --- as we please
for which the same is tru.

The proof is similar in case b —= 0.