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strayer_p48

# strayer_p48 - 48 Chapter 2 Con gruences 27 LetmeZwithm> 0...

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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 ...
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