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Let min EN 7 atZ be such theut Ca,m)=1, Ca,n)=| ¥ min)=\, Let K = Om (a) I j= On(a). 31K that ak=I (mod mn), ) Suppove -Prove

Let min EN 7 atZ be such theut Ca,m)=1, Ca,n)=| ¥ min)=\, Let K = Om (a) I j= On(a). 31K that ak=I (mod mn), ) Suppove -Prove

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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Let min EN 7 atZ be such that ca,m)=\, Capn)= | ģ (min) =\, Let K= Om(a) ģ j =OnCa). %D 51K Suppore -Prove that ak =I (mod mn), (1)
Transcribed Image Text:Let min EN 7 atZ be such that ca,m)=\, Capn)= | ģ (min) =\, Let K= Om(a) ģ j =OnCa). %D 51K Suppore -Prove that ak =I (mod mn), (1)
Theorem 1 (Fermat's Little Theorem). Let p be prime. If pła, then aP-l =1 (mod p). Corollary 2. Let p be a prime. Then for any integer a we have aP = a (mod p). Theorem 3 (Wilson's Theorem). If p is a prime, then (p – 1)! = -1 (mod p). Theorem 4. Let a, b e Z and m, k E N. If a = b (mod m) and k| m, then a = b (mod k). Theorem 5 (Cancellation). Let m > 1. If (a, m) = 1 and ab = ac (mod m), then b = c (mod m). Theorem 6. Consider the linear congruence equation ax = b (mod m) (1) in the unknown x. Let d = (a, m). Then equation (1) has solutions if and only if d b. Moreover, if d|b, then the solution to (1) is unique (mod m) and if d > 1, then equation (1) has exactly d incongruent solutions (mod m). Corollary 7. Consider the congruence equation ax = b (mod m). If (a, m) = 1, then this equation has a solution and it is unique (mod m). = 1. The order (mod m) of a, Definition 8. Let m e N and let a e Z be such that (a, m) denoted by om(a), is the least d e N such that ad = 1 (mod m); that is, om(a) = d if and only if 1. ad = 1 (mod m), and 2. a' # 1 (mod m) for all i e N such that i < d. = 1. Let n be a natural number. Then Theorem 9. Let m e N and let a E Z be such that (a, m) a" = 1 (mod m) if and only if om(a) | n. Theorem 10. Let a, b, n e Z. If n and a are relatively prime and n (ab), then n b. Theorem 11. Let a, b, n be integers where a n and b|n. If (a, b) = 1, then (ab) | n. Lemma 12. Let a and b be natural numbers and let p be a prime. If p|(ab), then p |a or p|b. Corollary 13. Let a be a natural number and p be a prime. If p|a?, then p|a. Definition 14. A function f: N → N is said to be multiplicative if for all pairs of relatively prime natural numbers m and n, we have that f(m · n) = f(m) · f(n). Lemma 15. Let p be an odd prime number and let a, b e Z be such that pła and płb. Then (1) ) (2) If a = b (mod p), then (4) = (9). = 1. (3) () (4) ) = 1. Theorem 16. Let p be an odd prime number. Then (금) - (1, 1-1, if p = 3 (mod 4). if p = 1 (mod 4); =(-1) Theorem 17. Let p be an odd prime number. Then if p = 1,7 (mod 8); –1, if p = 3, 5 (mod 8). 1, = Theorem 18 (Quadratic Reciprocity Law). Let p and q be distinct odd primes. Then S1, (29 -(-1)5-{1, if p=q = 3 (mod 4). if p = 1 (mod 4) or q = 1 (mod 4); %3D Corollary 19. Let p and q be distinct odd primes. If p = 1 (mod 4) or q = 1 (mod 4), then (E) ) = 1 and thus, (2) = (3). If p = q = 3 (mod 4), then () () -(). = -1 and thus, (2)
Transcribed Image Text:Theorem 1 (Fermat's Little Theorem). Let p be prime. If pła, then aP-l =1 (mod p). Corollary 2. Let p be a prime. Then for any integer a we have aP = a (mod p). Theorem 3 (Wilson's Theorem). If p is a prime, then (p – 1)! = -1 (mod p). Theorem 4. Let a, b e Z and m, k E N. If a = b (mod m) and k| m, then a = b (mod k). Theorem 5 (Cancellation). Let m > 1. If (a, m) = 1 and ab = ac (mod m), then b = c (mod m). Theorem 6. Consider the linear congruence equation ax = b (mod m) (1) in the unknown x. Let d = (a, m). Then equation (1) has solutions if and only if d b. Moreover, if d|b, then the solution to (1) is unique (mod m) and if d > 1, then equation (1) has exactly d incongruent solutions (mod m). Corollary 7. Consider the congruence equation ax = b (mod m). If (a, m) = 1, then this equation has a solution and it is unique (mod m). = 1. The order (mod m) of a, Definition 8. Let m e N and let a e Z be such that (a, m) denoted by om(a), is the least d e N such that ad = 1 (mod m); that is, om(a) = d if and only if 1. ad = 1 (mod m), and 2. a' # 1 (mod m) for all i e N such that i < d. = 1. Let n be a natural number. Then Theorem 9. Let m e N and let a E Z be such that (a, m) a" = 1 (mod m) if and only if om(a) | n. Theorem 10. Let a, b, n e Z. If n and a are relatively prime and n (ab), then n b. Theorem 11. Let a, b, n be integers where a n and b|n. If (a, b) = 1, then (ab) | n. Lemma 12. Let a and b be natural numbers and let p be a prime. If p|(ab), then p |a or p|b. Corollary 13. Let a be a natural number and p be a prime. If p|a?, then p|a. Definition 14. A function f: N → N is said to be multiplicative if for all pairs of relatively prime natural numbers m and n, we have that f(m · n) = f(m) · f(n). Lemma 15. Let p be an odd prime number and let a, b e Z be such that pła and płb. Then (1) ) (2) If a = b (mod p), then (4) = (9). = 1. (3) () (4) ) = 1. Theorem 16. Let p be an odd prime number. Then (금) - (1, 1-1, if p = 3 (mod 4). if p = 1 (mod 4); =(-1) Theorem 17. Let p be an odd prime number. Then if p = 1,7 (mod 8); –1, if p = 3, 5 (mod 8). 1, = Theorem 18 (Quadratic Reciprocity Law). Let p and q be distinct odd primes. Then S1, (29 -(-1)5-{1, if p=q = 3 (mod 4). if p = 1 (mod 4) or q = 1 (mod 4); %3D Corollary 19. Let p and q be distinct odd primes. If p = 1 (mod 4) or q = 1 (mod 4), then (E) ) = 1 and thus, (2) = (3). If p = q = 3 (mod 4), then () () -(). = -1 and thus, (2)
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