(a) Does j ≡ k(mod n) imply a^j ≡ a^k(mod n) for all a ∈ Z? Justify your answer.(b) Does a ≡ b(mod n) imply a^i ≡ b^i(mod n) for all a, b ∈ Z, i ∈ Z+? Justify your answer.

Q: Is there a solution to x^2 - 9x + 17 = 0 mod 73?

A: The given in this question is a quadratic equation in the form of x2-9x+17=0, and the objective is…

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

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: Prove by induction E 3₁ = (3^+¹ -1) 2 i=o

A:

Q: why are the polynomials x^4 + 2x^2 + x + 1 and x^4 + 1 are not irreducible over Z3 (mod3)

A:

Q: Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

A:

Q: Prove that if gcd(a, m) =1 and gcd(a – 1, m) =1, then 1 + a + a^2 + ....+ a^ phi(m)-1 congruent to 0…

A: Given gcda,m=1 and gcda−1,m=1 We have to show that 1+a+a2+......+aϕm−1≡0modm Since, gcda,m=1⇒m/a And…

Q: Let & € N. Prove by using the contra-positive that if ³ is odd then is odd. Proof: Choose which of…

A: Introduction: The basis for proof by contra positive is the idea that an implication equals its…

Q: Consider the points which satisfy the equation   y^2≡ x^3 + ax + b mod p where a=2, b=9 , and…

A: The given equation is,

Q: Prove: If a and b are relatively prime integers, then a^(theta(b)) + b^(theta(a)) congruent 1(mod…

A:

Q: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers a…

Q: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

Q: If a≡3 (mod 11) and b≡9 (mod 11): (a) Compute a+b(mod 11) (b) Compute a^3(mod 11)

A:

Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

A:

Q: How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I…

A:

Q: show that 2 is a primitive root mod 3^e for all e>=1

A:

Q: Use Euler's Theorem to find a number x, 0 <x<7, such that x^21 = 3 (mod 8)

A: By Euler's theorem, for any number x such that 0 &lt; x ≤ 7,        xφ(8) ≡ 1 (mod 8)       =&gt; x4…

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

A: Prove it considering two cases.

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: a) Find a0, a1, a2, a3 so that 1234 = a3 · 10^3 + a2 · 10^2 + a1 · 10^1 + a0 · 10^0 b) Prove that…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: is the following fucntion analytic in mod(z)&lt;2 and/or mod(z)&gt;2. sqrt(z^(2)+2z+2)/ z^(2)-4z+3

A: The objective of the question is to determine whether the given function is analytic in the regions…

Q: Show that (-3) = 1 if q = ±1 mod 12 -1 if q = ±5 mod 12

A:

Q: Find the smallest positive integer k satisfying the stated condition or explain no such k exists:…

A: To find the smallest positive integer k satisfying the given condition of congruence.

Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

A: I have proved it by using defination of congruent.

Q: Let p=17. If a belongs to F_{17} and a is non zero , compute the value of a^{16}(mod 17) for a=3 .

A: