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5 (9 9 () 2. Find a reduced residue system modulo 2", where m is a positive integer. in a reduced residue system modulo m, where m is a positive

5 (9 9 () 2. Find a reduced residue system modulo 2", where m is a positive integer. in a reduced residue system modulo m, where m is a positive

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Using the Euler's Theorem how can I solve 6.3 Question 2

1 (mod m). 11. Solve each of theorem. 12. Solve of the linear using theorem. llary 4.5.1, we can conclude that a" We can use Euler's theorem to find inverses modulo m. If a and m are relatively prime, we know that a . q®(m)–1 - a»(m) = 1 (mod m). Hence, a$(m)-lis an inverse of a modulo m. (u)4D = Example 6.20. We know that 2º(9)–1 – 26–1 – 25 = 32 = 5 (mod 9) is an inverse of 2 modulo 9. %3D We can solve linear congruences using this observation. To solve ax =b (mod m), where (a, m) = 1, we multiply both sides of this congruence by aº(m)-1 to obtain ax = a$(m)-1b (mod m). | | Therefore, the solutions are those integers x such that x = aº(m)-lb (mod m). | Example 6.21. The solutions of 3x = 7 (mod 10) are given by x = 3%(10)-1.7= 33.7=9 (mod 10), because ø(10) = 4. 6.3 EXERCISES 1. Find a reduced residue system modulo each of the following integers. a) 6 b) 9 c) 10 d) 14 e) 16 f) 17 2. Find a reduced residue system modulo 2", where m is a positive integer. 3. Show that if c1, c2, . . . , C6(m) is a reduced residue system modulo m, where m is a positive integer with m # 2, then c1 + c2 + · + Có (m) = 0 (mod m). .. 4. Show that if a and m are positive integers with (a, m) = (a – 1, m) = 1, then 1+ a + a² + .+a®(m)-1=0 (mod m). || ш 5. Find the last digit of the decimal expansion of 31000. 6. Find the last digit of the decimal expansion of 7999,999, 7. Use Euler's theorem to find the least positive residue of 3100,000 modulo 35. 8. Show that if a is an integer such that a is not divisible by 3 or such that a is divisible by 9, then a' = a (mod 63). 12 = 1 (mod 32,760). Show that if a is an integer relatively prime to 32,760, then a'² 10J Show that a$(b) + bø(a) = 1 (mod ab), if a and b are relatively prime positive integers. 1. Solve each of the following linear congruences using Euler's theorem. a) Sx = 3 (mod 14) b) 4x = 7 (mod 15) c) 3x = 5 (mod 16) 14. Solve each of the following linear congruences using Euler's theorem. a) 3x = 11 (mod 20) b) 10x = 19 (mod 21) c) 8x = 13 (mod 22) uppose that n = Pip2:.. p, where pj, P2, . . . , Pk are distinct odd primes. Show that
Transcribed Image Text:1 (mod m). 11. Solve each of theorem. 12. Solve of the linear using theorem. llary 4.5.1, we can conclude that a" We can use Euler's theorem to find inverses modulo m. If a and m are relatively prime, we know that a . q®(m)–1 - a»(m) = 1 (mod m). Hence, a$(m)-lis an inverse of a modulo m. (u)4D = Example 6.20. We know that 2º(9)–1 – 26–1 – 25 = 32 = 5 (mod 9) is an inverse of 2 modulo 9. %3D We can solve linear congruences using this observation. To solve ax =b (mod m), where (a, m) = 1, we multiply both sides of this congruence by aº(m)-1 to obtain ax = a$(m)-1b (mod m). | | Therefore, the solutions are those integers x such that x = aº(m)-lb (mod m). | Example 6.21. The solutions of 3x = 7 (mod 10) are given by x = 3%(10)-1.7= 33.7=9 (mod 10), because ø(10) = 4. 6.3 EXERCISES 1. Find a reduced residue system modulo each of the following integers. a) 6 b) 9 c) 10 d) 14 e) 16 f) 17 2. Find a reduced residue system modulo 2", where m is a positive integer. 3. Show that if c1, c2, . . . , C6(m) is a reduced residue system modulo m, where m is a positive integer with m # 2, then c1 + c2 + · + Có (m) = 0 (mod m). .. 4. Show that if a and m are positive integers with (a, m) = (a – 1, m) = 1, then 1+ a + a² + .+a®(m)-1=0 (mod m). || ш 5. Find the last digit of the decimal expansion of 31000. 6. Find the last digit of the decimal expansion of 7999,999, 7. Use Euler's theorem to find the least positive residue of 3100,000 modulo 35. 8. Show that if a is an integer such that a is not divisible by 3 or such that a is divisible by 9, then a' = a (mod 63). 12 = 1 (mod 32,760). Show that if a is an integer relatively prime to 32,760, then a'² 10J Show that a$(b) + bø(a) = 1 (mod ab), if a and b are relatively prime positive integers. 1. Solve each of the following linear congruences using Euler's theorem. a) Sx = 3 (mod 14) b) 4x = 7 (mod 15) c) 3x = 5 (mod 16) 14. Solve each of the following linear congruences using Euler's theorem. a) 3x = 11 (mod 20) b) 10x = 19 (mod 21) c) 8x = 13 (mod 22) uppose that n = Pip2:.. p, where pj, P2, . . . , Pk are distinct odd primes. Show that
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