CONGRUENCE 105

248. For example, in § 239 we had ¢ = 1-a + 0-b
and b = 0-a + 1-b
Instead we might have taken ¢ = (1 — bx)a + (0 + ax)b
and b = (0 — by)a + (1 + ay)d
In particular, taking @ = 15 and b = 6,
15 = (1 — 6x)15 + (0 + 15x)6

6 = (0 — 6y)15 + (1 + 159)6

Puttingx = 1 and y = — 1, we get
15 = (— 5)15 4 (15)6
6 = (6)15 4+ (— 14)6

Thus our table (§ 241) for finding 15 (X 6 and the numbers
I and m may be as follows:

 

 

 

Here (—17) X 15443 X 6 =3 =156

249. Th. [If a and b are two integers, not both zero, an infinit
number of pairs of integers, | and m, can be found, such that
la +mb = a®b.

This follows immediately from §§ 245, 247.

For example, (2 —452) X 24+ (—1+242) X45=24(Q45= 3

In particular, putting 2 = — 1,47 X 24 4+ (— 25) X45=3
puttingz =1, (—43)X24+4+23X45=3

250. Th. If a and b are two integers prime to each other, an
infinit number of pairs of integers, Il and m, can be found, such
that

la + mb = 1

For example, (—3 —72) X9+ (44+92) X7 =1
In particular, puttingz = — 2,11 X9 4 (—14) X7 =1