Prove that for any integers m,n where n > 0 there exists a unique integer 0 ⩽ r < n such that m ≡ r mod n. Since this integer r is unique, we will denote it by m modn

Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

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Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

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Q: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

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Q: Let x E Z. Prove that either x2 = 0 (mod 4) or x2 = 1 (mod 4).

A: By the help of even and odd integer's we prove the result. The answer is given as follows :

Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

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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: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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Q: Let p be an odd prime and a, b E Z such that (a, p) 1 and (b, p) = 1. Prove that if neither x² = a…

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Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: prove, for all integers x, if x mod 3 does not equal 0, then 3 | (x2 +2)

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Q: Let p be prime and a an integer not divisible by p. Prove that if a²"-1 (mod p), then a has order…

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Q: x

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Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

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Q: let a,b,c,d, and n be integers with n>1. Prove that if a is congruent to b(mod n) and c is congruent…

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Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: Find all values of m with 0 ≤ m ≤ 23 such that 10x ≡ m (mod 24) has a solution

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Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

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Q: 2) Find the smallest positive integer z, such that z = 13 mod 4 z = 20 mod 9 z = 11 mod 25 .

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Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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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: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

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

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