Let x be an integer. Prove that if x mod 3 is not 1, then 3|(x2+x).

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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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 x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

A: Given: x mod 9=5 then, 9|x2-2x+3|

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: Suppose a E Z. Prove that if a³ 1 (mod 5), then a # 1 (mod 5).

A: In this question, we use the method of contradiction to prove the desired result.

Q: Let n≥1, x be a real number, and x ≥ −1. Prove the following statement using mathematical induction.…

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Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

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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: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

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

A: I hove used the theorem on congruent

Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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Q: Prove that if a and b are integers, then (a^5 + b^5) is congruent to a^5 + b^5 modulo 5.

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Q: Let be a prime such that p = 1 (mod 4). Prove that –1F, is a square in Fp. -

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Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

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

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Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

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Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

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Q: Show that for any positive integer x, at least one of x, x+ 2, or x+ 4 is divisible by 3

A: The objective is to show that for any positive integer x at one of x, x+ 2, or x+ 4 is divisible by…

Q: Show that if x is an integer and x3 + 11 is odd, then x is even using a proof by contraposition.

A: We need to prove that if x is an integer and x3+11 is odd , then x is even by contraposition. The…

Q: Let x be an integer. Prove that, if xmod15 = 4, then 15|(x2-1)

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Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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Q: The congruence x² = 2(mod7) is not solvable: O False O True

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Q: Prove that xk = x* mod |G| for x € G for any integer k.

A: Prove it considering two cases.

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: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

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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: (1 + x)" > 1 + nx

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Q: Show that if a, b, and m are integers such that m > 2 and a = b (mod m), then ged(a, m) = gcd(b, m).

A: This question is related to Number Theory

Q: (b) Let x be an integer such that 0 < x < 8 and (10500 + 105000 ) = x (mod 9). Justifying your…

A: We have to find the required value of x .

Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: 1. Use Euler's Theorem to prove a = a (mod 105) for all a E Z.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let n E Z. Prove that 3n # 1 (mod 9).

A: Proof by contradiction: Direct approaches can be difficult (or impossible) to use to prove a…

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

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Q: Prove that the following two sets are equal by showing that each is a subset of the other: {14x + 6y…

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Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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Q: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: Prove that if p is a prime number then Z mod p is an integral domain

A: Integral domain:- A commutative ring R with a unit element 1 with no zero divisors is said to be…

Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

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Q: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

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

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…

Q: Let x be an integer. a) Prove that, if xmod15 = 7, then 3|(x2-1). b) Prove that, if xmod15 = 7,…

A: See the detailed solution below.

Q: Prove, for all integers x, if x mod 3 is not 0, then 3|(x2+2).

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let n be any fixed positive integer. Then…

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: Prove that whenever x ∈Zand d ∈Z+, if d does not divide (x2 −3x), then x mod d does not divide 3

A: The given statement is False.  Consider, x = 5 and d = 4. Then we see, x2 - 3x = 52 - 3x5…

Q: Prove that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.

Q: Let n be a positive integer, and assume that a and b are integers such that a ≡ b (mod n). Prove…

A: Looking at the definition of congurence we can easily do these two problems and we need some algebra…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.