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A. Observe that 15 = 1 ??? 7 15^2 = 225 ???7 = 1 ??? 7 15^3 = 3375 ???7 = 1 ??? 7 Can you see a pattern? Using what you’ve learned from above, what is 15^289 ??? 7? B. Suppose ? = 5⨀? ۩ 3 in ?6.  Solve for x.

A. Observe that 15 = 1 ??? 7 15^2 = 225 ???7 = 1 ??? 7 15^3 = 3375 ???7 = 1 ??? 7 Can you see a pattern? Using what you’ve learned from above, what is 15^289 ??? 7? B. Suppose ? = 5⨀? ۩ 3 in ?6.  Solve for x.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 51E
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A. Observe that 15 = 1 ??? 7
15^2 = 225 ???7 = 1 ??? 7
15^3 = 3375 ???7 = 1 ??? 7
Can you see a pattern? Using what you’ve learned from above, what is 15^289 ??? 7?


B. Suppose ? = 5⨀? ۩ 3 in ?6. 
Solve for x.

Examples: 4 @ 5 — 9 тod7 — 2 тod7 5 03 — 8 тod7 — 1 mod7 ЗО3%3D 9 тоd7 — 2 тod7 6043 24 тod6 — 3 тod7 e 012 3 4 5 6 00 12 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 45 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 12 3 5 5 601 2 3 4 6 6 012 3 4 5 O 01 2|3 4 5 6 In Z, , the additive inverses are 0 00000 0 0 1012 3 4 5 6 2 0 2 46 13 5 3 036 2 5 1 4 4 04 15 2 6 3 5 053 16 4 2 6 06 5 4 3 2 1 1 2 4 5 Inv 5 4 3 2 1 Since 7 is prime , all nonzero elements have mult. inverse 101 = 1 mod7 2 O 4 = 1 mod7 305%3D 1 тоd7 4 © 2 = 1 mod7 5 0 3 = 1 mod7 606=1 mod7 Addition and Multiplication in Z, 1 2 3 4 6 Mult 4 6. Inv
Transcribed Image Text:Examples: 4 @ 5 — 9 тod7 — 2 тod7 5 03 — 8 тod7 — 1 mod7 ЗО3%3D 9 тоd7 — 2 тod7 6043 24 тod6 — 3 тod7 e 012 3 4 5 6 00 12 3 4 5 6 1 1 2 3 4 5 6 0 2 2 3 45 6 0 1 3 3 4 5 6 0 1 2 4 4 5 6 0 12 3 5 5 601 2 3 4 6 6 012 3 4 5 O 01 2|3 4 5 6 In Z, , the additive inverses are 0 00000 0 0 1012 3 4 5 6 2 0 2 46 13 5 3 036 2 5 1 4 4 04 15 2 6 3 5 053 16 4 2 6 06 5 4 3 2 1 1 2 4 5 Inv 5 4 3 2 1 Since 7 is prime , all nonzero elements have mult. inverse 101 = 1 mod7 2 O 4 = 1 mod7 305%3D 1 тоd7 4 © 2 = 1 mod7 5 0 3 = 1 mod7 606=1 mod7 Addition and Multiplication in Z, 1 2 3 4 6 Mult 4 6. Inv
Examples: 3Ө5 3 8 тod6 — 2 тod6 2 0 4 = 6 mod6 = 0 mod6 4 0 3 = 7 mod6 = 1 mod6 2 О33D 6тod6 — 0 тod6 4 05 %3D 20 тod6 — 2 тоd6 Review O0 12 3 00 0000 0 10 12 3 4 5 2 0 240 2 4 3 03 O 01 2 34 5 00 12 3 4 5 1 1 2 3 4 5 0 2 2 3 45 0 1 3 34 5 01 2 4 45 01 2 3 5 5 01 In Z6 , the additive inverses are 2 4 5 3 0 3 0 4 04 2 0 4 2 5 0 5 4 3 2 Inv. 4 1 2 3 4 1 Since 6 is not prime and k 1 2 3 4 5 Gcd(k,6) 1 1 Addition and Multiplication in Z6 Only 1 and 5 have multiplicative inverse 1013D1 тod6 5 05%3 1 тоd6 k 1 5 | Inv.(reciprocal) 1
Transcribed Image Text:Examples: 3Ө5 3 8 тod6 — 2 тod6 2 0 4 = 6 mod6 = 0 mod6 4 0 3 = 7 mod6 = 1 mod6 2 О33D 6тod6 — 0 тod6 4 05 %3D 20 тod6 — 2 тоd6 Review O0 12 3 00 0000 0 10 12 3 4 5 2 0 240 2 4 3 03 O 01 2 34 5 00 12 3 4 5 1 1 2 3 4 5 0 2 2 3 45 0 1 3 34 5 01 2 4 45 01 2 3 5 5 01 In Z6 , the additive inverses are 2 4 5 3 0 3 0 4 04 2 0 4 2 5 0 5 4 3 2 Inv. 4 1 2 3 4 1 Since 6 is not prime and k 1 2 3 4 5 Gcd(k,6) 1 1 Addition and Multiplication in Z6 Only 1 and 5 have multiplicative inverse 1013D1 тod6 5 05%3 1 тоd6 k 1 5 | Inv.(reciprocal) 1
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