162 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

403. Def. Dividing by any such common factor, we get an
equivalent equation, one equation being called equivalent to
a second when their solutions are the same.

Thus 2x 4 6y = 8 is equivalent to x 4+ 3y = 4.

404. Dividing by the greatest common factor of ¢ and b,
the new coefficients of x and y will be prime to each other.
§ 235.
So we need to consider only the equation ax + by = ¢ in
which a, b, ¢ are integers, a == 0, b—= 0, and « || .
From this point on we will therfor assume that these con-
ditions are satisfied.

405. Th. One solution of the equation can always be found,
namely (lc, mc), where I and m are two integers such that

al + bm = 1. § 246.
For a(lc) + b(mc) = (al + bm)c = c.
Example. 49x + 19y = 900

Here ¢ = 49, b = 19, ¢ = 900.
To find a pair of values of / and m such that al + bm = 1
we proceed as in § 241.

 

 

7X49 + (—18) X 19 = 1

 

Wefind 49X 7+19 X (—18) =1
l=T7 m= —18
lc = 6300, mc = — 16200
Hence (6300, — 16200) is a solution of the given equation.

406. Th. If (x, y) and (x1, vi) are two solutions, then
x = x1 (mod b), y = y1 (mod a), and an integer n can be found,
such that x = x, + nb and y = y, — na.