Unformatted text preview: 48 Chapter 2 Con gruences 27. LetmeZwithm > 0. Prove that
{m—l mE3 m—3m—1} 2 1 12...m—
2 J 2 r r v ’0!!! y 2 ) 2 is a complete residue system modulo m if and only if m is odd. —___ Linear Congruences in One Variable We turn now to solving special types of congruences known as linear
congruences in one variable. Deﬁnition 4: Let a, b 62. A congruence of the form ax E b modm is
. said to be a linear congruence in the variable x. The congruence is linear in the sense that the variable x occurs to the ﬁrst
power; more general congruences will be studied in later chapters. In a linear
congruence in one variable, we are interested in ﬁnding all integer values of
the variable for which the congruence is true. We illustrate with several
examples. Example 5: (a) The congruence 2x E 3 mod4 is a linear congruence in the variable x.
Modulo 4, there are only four values that x can attain, namely, 0, 1, 2, and 3.
(See Example 4.) A quick check shows that none of these values substituted
for x in 2x E 3 mod 4 results in a true congruence; so the congruence
2x E 3 mod 4 is not solvable. (b) The congruence 2x E 3 mod5 is a linear congruence in the variable x.
Modulo 5, there are only ﬁve values that x can attain, namely, 0, 1, 2, 3, and
4. (See Example 4.) A quick check shows that only x E 4 results in a true
congruence; so the solution set for this congruence is 4 as well as any integer
congruent to 4 modulo 5. In other words, the solution set for this
congruence is {. . . , —6, —1, 4, 9, . . .}. Even though there are inﬁnitely many
solutions to 2x E 3 mod 5, there is exactly one incongruent solution to the
congruence modulo 5, namely, 4. (c) The congruence 2x E 4 mod 6 is a linear congruence in the variable x. It
is left to the reader to show that this congruence has inﬁnitely many
solutions but exactly two incongruent solutions modulo 6, namely, 2 and 5. (d) The congruence 3x E 9 mod 6 is a linear congruence in the variable x. It
is left to the reader to show that this congruence has inﬁnitely many
solutions but exactly three incongruent solutions modulo 6, namely, 1, 3,
and 5. As motivated in (b) of Example 5, if one element of a congruence class is a
solution of ax E b mod m, then all elements of the congruence class are ...
View
Full Document
 Spring '10
 BOYLAND

Click to edit the document details