Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence mod m to do this.

Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: Find all solutions for x, up to congruence. x ≡ 1 mod 4 and x ≡ 7 mod 13.

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Q: (b) Prove that if a = b (mod m) and if b = c (mod m), then a = c (mod m).

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Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: y(5-cos xey-y²) 2.xy +1 sin x dy 5-c -cos xe sin x d. xy + In y +ein x = 5x [Answer: dx or 2xy +- y

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Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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Q: Prove that if a = b (mod m) and if c = d (mod m), then a+c=b+d (mod m).

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Q: solve the following system of linear congruences x ≡ 2(mod 14) x ≡ 16(mod 21) x ≡ 10(mod 30)

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Q: #5. Suppose that a = 2 (mod 113), and b = 5 (mod 113) What is (a b + 56b + 155) mod 113?

A: given a≡2(mod113),b=5(mod113)a3≡23(mod113)a3b≡235(mod113)56b≡280(mod113)56b≡54(mod113)155≡15mod113)

Q: Prove that if a = b (mod m) and if c = d (mod m), then ac = bd (mod m).

A: a≡b   (mod m)⇒ m|(a−b)That is there exist an integer k such that a−b=mk

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: Determine all solutions x ∈ Z to the triple x ≡ 5 (mod 6), x ≡ 7 (mod 10) and x ≡ 2 (mod 15) of…

A: We reduce the congruence and then solve

Q: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

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Q: If gcd(m,n) = 1 is a given condition, how can you prove that the congruences x ≡ a (mod m) and x ≡ b…

A: The congruences have a solution when .

Q: Find all possible positive values of m: 47 = m mod (5)

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Q: Consider the congruence a² + x + 1 = 0 mod 21. Solutions to the congruence above must also be…

A: The given congruence equation is .We will solve this congruence relation using the Chinese remainder…

Q: 4. A linear code is given by the encoding function y = ax+b (mod 26) and the decoding function x =…

A: 4.(a) Given a = 11, b = 2, the encoding function is given by         y ≡ 11x + 2 (mod 26)…