Second picture is just theorems be such that Ca,m)=1,Let thE N and a,be ZCb ,m) = ), and Cab, m)=\. Let j= Om Ca);k= Om(b) and n= Om Cab), Suppose that(j,k) =d >I,for(abi%3DsoJこdrksdssomeProve thatnI rdsthen showthat Om Cab) < OmCa) Om (b). 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 equationax = 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 dincongruent solutions (mod m).Corollary 7. Consider the congruence equation ax = b (mod m). If (a, m) = 1, then this equationhas 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 if1. ad = 1 (mod m), and2. a' # 1 (mod m) for all i e N such that i < d.= 1. Let n be a natural number. ThenTheorem 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 relativelyprime 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. Thenif 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. ThenS1,(29 -(-1)5-{1, if p=q = 3 (mod 4).if p = 1 (mod 4) or q = 1 (mod 4);%3DCorollary 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)

Name: Second picture is just theorems be such that Ca,m)=1,Let thE N and a,b
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Description: Second picture is just theorems be such that Ca,m)=1,Let thE N and a,be ZCb ,m) = ), and Cab, m)=\. Let j= Om Ca);k= Om(b) and n= Om Cab), Suppose that(j,k) =d >I,for(abi%3DsoJこdrksdssomeProve thatnI rdsthen showthat Om Cab) < OmCa) Om (b). Theorem 1

Let m,n∈ℕ and a,b∈ℤ such that a,m=1, b,m=1 and ab,m=1 Suppose j=Oma, k=Omb and n=Omab Again, let…

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be such that Ca,m)=1, Let thE N and a,be Z Cb ,m) = ), and Cab, m)=\. Let j= Om Ca); k= Om(b) and n= Om Cab), Suppose that (j,k) =d >I, for (abi %3D so Jこdr ksds some Prove that nI rds then show that Om Cab) < OmCa) Om (b).

be such that Ca,m)=1, Let thE N and a,be Z Cb ,m) = ), and Cab, m)=\. Let j= Om Ca); k= Om(b) and n= Om Cab), Suppose that (j,k) =d >I, for (abi %3D so Jこdr ksds some Prove that nI rds then show that Om Cab) < OmCa) Om (b).

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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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Question

Second picture is just theorems

be such that Ca,m)=1,
Let thE N and a,be Z
Cb ,m) = ), and Cab, m)=\. Let j= Om Ca);
k= Om(b) and n= Om Cab), Suppose that
(j,k) =d >I,
for
(abi
%3D
so
Jこdr
ksds
some
Prove that
nI rds
then show
that Om Cab) < OmCa) Om (b).

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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