prove if a=b mod n then (a^k)=(b^k) mod n
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Problem Set 4 Solutions
22 févr. 2005 Prove that there exists an integer k?1 such that. k k?1 ? 1 (mod n). · provided gcd(k n)=1. Assume n > 1. Solution. If gcd(k |
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Congruences and Modular Arithmetic
Let n ? N. Then congruence modulo n is an equivalence relation on Z. Proof (Sketch). Let ab |
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Congruences
If n is an integer then a is congruent to b modulo n if and only if a and b have the same remainder when divided by n. Proof. By the division algorithm |
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Congruences (Part 1)
3) If a ? b (mod m) and n ? N then an = bn (mod m). Proof: Use Mathematical Induction. Case n = 1 True. Assume true for n = k. |
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3 Congruence
It replaces the con- gruence sign with an equality. Theorem 3.3 If a ? b mod n then b = a + nq for some integer q and conversely. Proof: If a |
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Number Theory
If a ? b (mod m) then ak ? bk (mod m) for any integer k Theorem: An integer n is divisible by 11 i the di erence of the sums of the odd. |
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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 for some k ? Z. Now by the Division Algorithm a and b can be written ... |
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3 Congruence
It replaces the con- gruence sign with an equality. Theorem 3.3 If a ? b mod n then b = a + nq for some integer q and conversely. Proof: If a |
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Congruence and Congruence Classes
Two congruence classes modulo n are either disjoint or identical. Proof. If [a]n and [b]n are disjoint there is nothing to prove. Suppose then that [a] |
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Congruences
Proof. (a) If a ? Z then a ? a =0=0 · n |
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3 Congruence - New York University
Theorem 3 3 If a b mod n then b = a+nq for some integer q and conversely Proof: If a b mod n then by de nition nj(b?a) Therefore b?a = nq for some q Thus b = a+nq Converselyifb = a+nqthenb?a = nq and so nj(b?a) and hence a b mod n then b = a+nq 25 |
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35: The Division Algorithm and Congruence - Mathematics LibreTexts
[a] = fx 2Zjx a(mod n)g: Note that 1 For any integer a [a] is a set not an integer 2 If 0 a < n then [a] can be described as the set of integers that give a reminder of a when divided by n (In this case we call a a standard representative of [a]:) 3 If [a] = [b] it does not mean a = b only that a b(mod n) or that a and b give the same |
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55: Congruences
[a] = fx 2Zjx a(mod n)g: Note that 1 For any integer a [a] is a set not an integer 2 If 0 a < n then [a] can be described as the set of integers that give a remainder of a when divided by n (In this case we call a a standard representative of [a]:) 3 If [a] = [b] it does not mean a = b only that a b(mod n) or that a and b give the |
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Homework 4 Solutions - University of California Berkeley
Prove that if a b (mod m) then a mod m = b mod m Simplest proof: If a b (mod m) then mja b so a = b+ mk for some k 2Z Then if a = qm+ r with 0 r < m it follows that b + km = qm + r and so b = (q k)m + r This implies that a mod m = b mod m Second proof: Let a = sm + r 1 and let b = tm + r 2 with 0 r 1;r 2 < m If a b (mod m) then |
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Congruences and Modular Arithmetic - Trinity University
Let nk ? Nand ab ? Z Then a ? b (mod n) ? ak ? bk (mod nk) Proof If a ? b (mod n) then a ?b = n?for some ?? Z Multiplying through by k yields ak ?bk = nk? so that ak ? bk (mod nk) Conversely if ak ? bk (mod nk) then ak ?bk = nk?for some ?? Z That is k(a ?b) = kn? |
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Searches related to prove if a=b mod n then a^k=b^k mod n filetype:pdf
Since a b( mod m) and c d( mod m) by the Theorem above there are integers s and t with b = a +sm and d = c +tm Therefore b +d = (a +sm)+(c +tm) = (a +c)+m(s +t) and bd = (a +sm)(c +tm) = ac +m(at +cs +stm) Hence a +c b +d( mod m) and ac bd( mod m) Corollary Let m be a positive integer and let a and b be integers Then (a +b) mod m = ((a |
How do you prove that a ? a (mod n)?
- This proves that n divides a ? a and hence, by the definition of congruence modulo n, we have proven that a ? a (mod n ). To prove the transitive property, we let n ? N, and let a, b, and c be integers.
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.
Is a B MODN a proof?
- Yes. That's a perfect proof. It's a matter of style a personal comprehension what is more convincing but when I think of what these mean I find a ? b modn; m | n ? a = b + kn = b + k a ? mod hits my in the gut just slightly harder, but that is just me.
How do you prove that n=k+1 is true?
- Use the fact that n=k is true, prove that n=k+1 is true. This shows that for some n=k, n=k+1 is true. By induction this means values are true. 4. Make a statement to conclude this in the structure:It has been prove for n=m, and for some n=k, n=k+1 is true.
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Congruences and Modular Arithmetic - Trinity University
Let n ∈ N and a,b ∈ Z We say that a is congruent to b modulo n, denoted a ≡ b 2 ac ≡ bd (mod n) Proof Write a − b = nk and c − d = nℓ with k,ℓ ∈ Z Then |
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Number Theory
If ka ≡ kb (mod m) and gcd(k,m) = 1, then a ≡ b (mod m) a ≡ b (mod m) i ak Prove that, for every k ⩾ 1, the rst, second, and third person executed cannot be 10, k, and k +1, in this Hence, n ≡ ak (−1)k +ak−1(−1)k−1 + −a1 +a0 = |
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Congruences - Mathtorontoedu - University of Toronto
3) If a ≡ b (mod m) and n ∈ N then an = bn (mod m) Proof: Use Mathematical Induction Case n = 1 True Assume true for n = k Induction Hypothesis: ak ≡ bk |
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Congruence and Congruence Classes
Proof (i) a − a = 0 and n 0, hence a ≡ a (mod n) (ii) a ≡ b (mod n) means that a − b = nk for some k ∈ Z Therefore, b − a = −nk = n(−k); hence b ≡ a (mod n ) [a]n, is the set of all integers that are congruent to a modulo n; i e , [a]n = {z |
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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 divided for some k ∈ Z Now by the Division Algorithm, a and b can be written |
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Congruences
(b) If a, b ∈ Z and a ≡ b (mod n), then a − b = k · n for some k ∈ Z, and so b − a Proof Suppose that a and b leave the same least nonnegative remainders |
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Math 371 Lecture §21: Congruence and Congruence Classes
(2) if a ≡ b (mod n), then b ≡ a (mod n) (congruence mod n is symmetric), and (2) ac ≡ bd (mod n) Proof We suppose that a − b = nk for some k ∈ Z and c |
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Homework 6 - Number Theory Homework
mod n Proof This is just an easy induction on k D Proposition 5 If f(x) = ckxk + ck-1xk-1 + ···c1x + c0 is a polynomial with integer coefficients, then a ≡ b mod n |
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CONGRUENCE AND MODULUS - CSUSM
let a, b, c, d ∈ Z If a ≡ b mod m and c ≡ d mod m then a + c ≡ b + d mod m Proof since 1 ≡ 1 mod m Assume that the statement holds for a particular n = k |
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