then a ≡ b mod mn
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Congruences
Intuitive idea : If two numbers a and b leave the same remainder when divided by a third number m then we say "a is congruent to b modulo m" and write a ≡ b |
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Basic Properties of Congruences
The notation a ≡ b (mod m) means that m divides a − b We then say that a is congruent to b modulo m 1 (Reflexive Property): a ≡ a (mod m) |
For a positive integer n, two integers a and b are said to be congruent modulo n (or a is congruent to b modulo n), if a and b have the same remainder when divided by n (or equivalently if a − b is divisible by n ).
It can be expressed as a ≡ b mod n.
What does a ≡ b mod m mean?
The notation a ≡ b (mod m) means that m divides a − b.
We then say that a is congruent to b modulo m.
1(Reflexive Property): a ≡ a (mod m) 2. (Symmetric Property): If a ≡ b (mod m), then b ≡ a (mod m).
How do you prove a ≡ b mod m?
To prove If a≡b mod m, then a2≡b2 mod m2
1If a≡b mod m, then ma−b i.e. a−b=km, k is an int.2) Square both sides we get: (a−b)2=k2m2⟹a2−2ab+b2=k2m2.
3) Then a2+b2=2ab+k2m2.
4) This means that m2a2+b2 and not m2a2−b2 i.e. a2≡b2 mod m2.
5) Therefore the statement If a≡b mod m, then a2≡b2 mod m2, is false.
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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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THE CHINESE REMAINDER THEOREM We should thank the
Then we will show it is unique modulo mn. Existence of Solution. To show that the simultaneous congruences x ? a mod m x ? b mod n. |
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3 Congruence
Theorem 3.2 For any integers a and b and positive integer n |
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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 by n. Proof. ?. Suppose a ? b (mod n). Then by definition |
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Contents 2 Modular Arithmetic in Z
If p(x) is any polynomial with integer coefficients then a ? b (mod m) a is a unit modulo mn if and only if it is a unit modulo m and modulo n. |
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Chapter 2 The ring Z/mZ
(iii) transitive: if a ? b (mod m) and b ? c (mod m) then a ? c. (mod m). relatively prime |
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Math 110 Homework 3 Solutions
Jan 29 2015 invertible both modulo n and modulo m. (c) Use the result of part (b) to show that ? is multiplicative: if gcd(m |
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The Chinese Remainder Theorem Theorem. Let m and n be two
two integers. Then the two congruences x ? a (mod m) x ? b (mod n) have common solutions. Any two common so- lutions are congruent modulo mn. |
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3 Congruence
Theorem 3 3 If a ≡ b mod n then b = a + nq for some integer q, and conversely Proof: If a ≡ b mod How many days before she reaches 9 minutes before 12? |
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2 Congruences and modular arithmetic The notation for congruence
(iii) Transitive: If a ≡ b (mod m) and b ≡ c (mod m) then a ≡ c (mod m) a ≡ b ( mod d), and the solution (when it exists) is unique modulo lcm(m, n) = mn/d |
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BASIC PROPERTIES OF CONGRUENCES The letters a, b, c, d, k
Then a ≡ b (mod mn) 11 Suppose that a ∈ Z Then there exists a unique integer r such that a ≡ r (mod m) and 0 |
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Number Theory
If a ≡ b (mod m), then a+um ≡ b +vm (mod m) for every integers u and v If ka ≡ kb (mod m) and gcd(k,m) = 1, then a ≡ b (mod m) a ≡ b (mod m) i ak ≡ bk |
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MA 261 — Worksheet Wednesday, April 16, 2014 1 Theorem 327
16 avr 2014 · Then the system x ≡ a (mod n) x ≡ b (mod m) has a unique solution modulo mn Proof: By Theorem 3 27 we know that the system has a |
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Congruences - Mathtorontoedu - University of Toronto
"a is congruent to b modulo m" means m(a-b) Equivalently, a b leave the same remainder by division by m (for a,b≥0) 1) If a ≡ b (mod m) then (a+c) ≡ (b+d) |
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Congruence and Congruence Classes
The next definition yields another example of an equivalence relation Definition 11 2 Let a, b, n ∈ Z with n > 0 Then a is congruent to b modulo n; a ≡ b (mod n) |
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Prove: For integers a, b and positive integer m, a ≡b (mod m) if
“both directions ”) First, assume a ≡b (mod m) Then m (a – b), so there is k ∈ Z such that a – b = mk Let a mod m = r Then, according to the division algorithm, |
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LECTURE 4: CHINESE REMAINDER THEOREM AND
then x is a solution of (1 1) if and only if x ≡ x0 (mod m1m2 mr) Proof Let m = m1m2 identically zero, and (b) whenever (m, n) = 1, one has f(mn) = f(m)f(n); |
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Contents 2 Modular Arithmetic in Z
If p(x) is any polynomial with integer coefficients, then a ≡ b (mod m) implies prime, we conclude that ad ≡ 1 (mod mn), meaning that a is a unit mod mn as |
