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

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

A:

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…

A:

Q: Given 6x + 2 ≡ 101 (mod 7), find an integer z such that 2x ≡ z (mod 7)

A:

Q: Find an integer x satisfying the equation: 4x21 = 2(mod 5) or show that no such integer exists.

A:

Q: Prove the following statement or disprove by giving a counterexample "for all positive intergers n…

A: We have to prove or disprove the following statement For all positive integer n and for all integer…

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

A:

Q: Show that the positive integers less than 11, except 1 and 10, can be split into pairs of integers…

A:

Q: Prove for any integers a and b, if (a3+b3) mod 3 is not 0, then (a+b) mod 3 is not 0. (Hint: recall…

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

Q: wing statement is true or false. provide a counterexample for each false statement.  x is even iff…

A: In this question, the concept of Modular Arithmetic is applied. Modular Arithmetic Modular…

Q: Use the table in VII to give a formal proof of the following. For all integers a and b, if a = 1…

A:

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

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

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

A:

Q: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

A:

Q: Prove that if p is a prime and a is any integer so that p does not divide a, then the additive order…

A:

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

A:

Q: 5. Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n). (Hint: use the…

A: The greatest common divisor (gcd) of two or more numbers is the greatest common factor number that…

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

A:

Q: Show that if a,b, and n are integers such that n≥2 and a≡b (mod n), then gcd(a, n) = gcd(b, n).

A:

Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

A:

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

A:

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

A:

Q: Let C={x €Z: x =7 mod 9} and let D={x € Z,x = 1 mod 3). a) List five elements of C and five elements…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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

A:

Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

A:

Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

A:

Q: 20. Are the following statements true or false? Either prove the statement is true or provide a…

A: PART A For the integers a and b, a.b=0(mod 6) This implies that either a=0(mod6) or b=0(mod6). So,…

Q: Prove the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7

A: We will prove the given statement.

Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

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

A:

Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: Let n be an integer. Prove that, if n mod6 = 2, then n2mod12=4

A: We apply concept of modulus here to solve this.

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Find the nonnegative integer a less than 28 represented by each of these pairs, where each pair…

A: According to the given information, it is required to find the non-negative integer a less than 28…

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

A:

Q: a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the…

A:

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

A:

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

A: Given x be an integer and xmod3≠1 then We have to prove that 3|x2+x Given xmod3≠1 This implies 3/x…

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: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

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.

A:

Q: (a) Consider the following statement: For all n € N, x(x − 1) = 0 mod n Which of "if", "only if",…

A:

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…