PDF show that if a and b are integers with a ≡ b mod n then f(a ≡ f(b mod n)) PDF



PDF,PPT,images:PDF show that if a and b are integers with a ≡ b mod n then f(a ≡ f(b mod n)) PDF Télécharger




[PDF] Number Theory

10 oct 2019 · a b Any integer n has divisors ±1 and ±n, called the trivial divisors of n If a is a divisor of n, (Not required) We first show that a has a factorization into primes If a has only the a ≡ b (mod d) (c) a ≡ b (mod n) iff b − a = kn for an integer k, which is iff b of a modulo p Then fb is the inverse function of fa
Number Theory


[PDF] Solutions to Homework Set 3 (Solutions to Homework Problems

a ≡ b (modn) D 2 1 2 If a ∈ Z, prove that a2 is not congruent to 2 modulo 4 or to 3 modulo 4 • Proof By the Division Algorithm any a ∈ Z must have one of the  
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[PDF] 88 Exercises - Cal State LA

Let a and b be any integers Let fa : Zn → Zn be given by fa(x) = a · x as in Exercise 8 18 (a) Prove that fab = fba (b) Prove: If a ≡ b(mod n), then fa = fb (2) Let f 
chapter kps






[PDF] Cryptography (Math 134) - Robert Boltje

This shows that rn is the greatest common divisor of a and b the proof 1 19 Definition (a) Two non-zero integers a and b are called coprime or (d) If a ≡ b mod m and a ≡ b mod n with gcd(m, n) = 1 then a ≡ b mod mn b = f−b We call b the key of this crypto system For b =3+27Z we obtain TOMORROW fb
f m notes


[PDF] Math 238: Solutions to Homework - Williams College

15 sept 2011 · and FbFn Assume a, b > 2, then Fa,Fb ≥ 2 and hence 1 < Fa,Fb < Fn In Find a pattern and prove it for the sum of the first n odd integers Solution: Note that x2 ≡ 0 mod 3 if x ≡ 0 mod 3, and x2 ≡ 1 mod 3 if x is either 1 
Math HWSolns


[PDF] Solutions Chapters 1–5 - Illinois

The computation of Problem 8 shows that the left cosets of H do not coincide with Ψ(ab) = fab = fa ◦ fb = Ψ(a)Ψ(b), so Ψ is a homomorphism (see the solution to If G is abelian, then fa(x) = axa if ab = c as integers, then ab ≡ c modulo n
SolutionsChap


The cardinality of the value sets modulo n of x2+x-2 - Project Euclid

Consider the modular circle a,n ={(x, y):x2+y2 ≡a (mod n), 0≤x, y ≤n−1} and the Figure 2 shows the two superimposed, and the intersection 1,997 ∩ 997 n (where n = /n is the set of integers ↓fa× fb n r −−−→ a × b commutes D Thus we have reduced the problem of determining formulas for #I(x2 +y2, n)
involve. . .






[PDF] Elementary Number Theory and Algebra - webusersimj-prgfr

numbers and divisibility, uniqueness of factorisation for integers, Euclid's 3 4 Invertible elements in Z/nZ, congruences ax ≡ b (mod n) 4 3 10 Computing the order of a (mod n) in (Z/nZ)∗ 1 1 2 Factorisation into primes Experience shows that positive integers admit fA ∈ K[X], called the minimal polynomial of A 127 
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[PDF] MA3H1 Topics in Number Theory Example Sheet 2 You should

(2) (a) Let N ≡ 3 (mod 4) be a positive integer Show that (b) Show that there are infinitely many primes p ≡ 3 (mod 4) (3) Let p (a) Show that if 2m + 1 is prime then m = 2n for some n (b) The Show that if a = b the gcd(Fa,Fb) = 1 (c) If p 
assign



3 Congruence

Theorem 3.3 If a ? b mod n then b = a + nq for some integer q and conversely. We can now show some useful algebraic properties of congruences.



Solutions to Homework Set 3 (Solutions to Homework Problems

Prove that a ? b (mod n) if and only if a and b leave the same remainder when Since the set of prime numbers in Z is infinite we can always find a ...



Number Theory

assumption these are all the primes but N is seen not to be divisible by any of If f is a polynomial with integer coefficients and a ? b (mod m)



3 Congruence

Theorem 3.3 If a ? b mod n then b = a + nq for some integer q and conversely. We can now show some useful algebraic properties of congruences.



Group Theory

To show that it indeed computes the greatest common divisor of a and b we Two integers a



CHAPTER 5 Contrapositive Proof

F. F. T. T. T. T. According to the table statements P ? Q and ? Q ?? P exercises for this chapter asks you to show that if a ? b (mod n)



CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and

look at leading coefficients to show that if f(X) = 0 and g(X) = 0 then (i) If a and b are integers that are congruent modulo n



UNIT – I –Theory of Divisibility and Congruencies– SMTA5203

Corollary If a and b are integers with ? 0



IIT Kharagpur 1 Basic Properties of Integers II

Our next propositions show a few properties of the solutions. Proposition 5. 1. If the linear congruence ax ? b(mod n) has a solution then there is a.



Solutions

Suppose A1 = {ab



3 Congruence - New York University

Theorem 3 2For any integers a and b and positive integer n we have: 1 a amodn 2 If a bmodn then b amodn 3 If a bmodn and b cmodn then a cmodn These results are classically called: 1 Re?exivity; 2 Symmetry; and 3 Transitivity The proofisasfollows: 1 nj(a? a) since 0 is divisible by any integer Thereforea amodn 2



Prove that $ ab mod n = (a mod n) · (b mod n)

Theorem 2 1For a positive integern and integersa; b; c we have a a(mod n) (congruence modnis re exive) ifa b(mod n) thenb a(mod n) (congruence modnis symmetric) and ifa b(mod n) andb c(mod n) thena c(mod n) (congruence modnistransitive) Remark Thus congruence modnis an equivalence relation onZ Proof



Chapter 2 The Ring of Integers Modulo N - Mathematics

For a;b 2Z we have a = b mod n if and only if a and b have the same remainder when divided by n In particular for every a 2Z there is a unique r 2Z with a = r mod n and 0 r < n Proof: Let a;b 2Z Use the Division Algorithm to write a = qn + r with 0 r < n and b = pn + s with 0 s < n We need to show that a = b mod n if and only if r = s



THE CHINESE REMAINDER THEOREM - University of Connecticut

To show that the simultaneous congruences amodm; x bmodn have a common solution inZ we give two proofs First proof: Write the rst congruence as an equation inZ sayx=a+myfor some 2Z Then the second congruence is the same as a+my bmodn: Subtractingafrom both sides we need to solve foryin (2 1) my b amodn:



Everything You Need to Know About Modular Arithmetic

Sum rule: IF a ? b(mod m) THEN a+c ? b+c(mod m) (3) Multiplication Rule: IF a ? b(mod m) and if c ? d(mod m) THEN ac ? bd(mod m) (4) De?nition An inverse to a modulo m is a integer b such that ab ? 1(mod m) (5) By de?nition (1) this means that ab ? 1 = k · m for some integer k As before there are may be many



Math 430 { Problem Set 2 Solutions - MIT Mathematics

3 22 Show that addition and multiplication mod nare well de ned operations That is show that the operations do not depend on the choice of the representative from the equivalence classes mod n Solution Suppose that a b(mod n) and c d(mod n) Then there are integers r;swith a= b+rn and c= d+ sn We nd that a+ c= b+ rn+ d+ sn = b+ d+ (r+ s)n;

What does (a + b) mod n mean?

( ( a + b) mod n) means that there is an integer k such that 0 ? a + b ? n k < n, and ( a mod n) + ( b mod n) means there are integers k 1 and k 2 such that 0 ? a ? n k 1 < n and 0 ? b ? n k 2 < n ? 0 ? a + b ? n ( k 1 + k 2) < 2 n which has no relation to the 0 ? a + b ? n k < n especially resulting same numbers to say they are equal.

Why is a mod b not a remainder?

It is because a mod b isn't simply the remainder as returned by the operator '%'. See some examples: There are some other definitions in math and other implementations in computer science according to the programming language and the computer hardware. Please see Modulo operation from Wikipedia.

Is (a + b) always an integer?

Let us say ‘a’ and ‘b’ are two integers, either positive or negative. When we add the two integers, their result would always be an integer, i.e (a + b) would always be an integer.

What is the answer to a*b (mod 10)?

So to find the answer to "A*B (mod 10)" all you need to do is find: "the one's place of A" and "the one's place of B" Similarly, if you instead used base N instead of base 10 to write out your numbers A and B, you'd find the exact same pattern.

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