(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: Prove if a=b mod m then a^n=b^n mod m use Math induction

A: We have to prove that if a=b mod m then a^n=b^n mod m

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: Let a, m € Z+ and (a,m) = (a - 1, m) = 1. Then prove that 1+a+...+ a(m)-1 = 0(modm)

A: For given,       (a,m) = (a -1, m) = 1 We have to show,      1 + a + a 2 + ........ + a φ (m)-1 ≡ 0…

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

A:

Q: If x = y (mod n) and u = v (mod n) then which of the following is false. (a) x+ u = y +v (mod n) (b)…

A: Given that x≡ymod n and u≡v mod n. By definition, x≡ymod n implies x-y is divisible by n. Then let,…

Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE.  Consider S ⊆ℤ, defined by  S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }.  Then the given…

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: 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: 1. Find all x, y, and z such that a) x = 3 (mod 4) x = 5 (mod 21) x = 7 (mod 25) b) 3y = 9 (mod 12)…

A:

Q: L-) Find all in tegers such thet 5x = 12 (mod 7) 10x = 4 (mod 11) E 16 (mod 10d ond 1occ < x < 3000

A: To determine: All the integers x such that 5x≡12 (mod 7)10x≡4(mod11)7x≡16(mod 10)and 1000≤x≤3000

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: If x = y (mod n) and u = v (mod n) then which of the following is true (a) x + u = y + v (mod n) (b)…

A: Given that x≡y (mod n) and u≡v (mod n)

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: 5. Explain why the congruence r = 2 (mod 4) has no solutions in Z.

A: We have to show that the congruence x2≡2(mod 4) has no solution in ℤ

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: Let k be the order of "a" in n mode, then prove that, o(ah) = k (h, k) = 1

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: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

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: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: てheorem' Let p be aprime number, a integer (ap)-1 0nd pacitive integer. In this cose; ) #1 (mod p)…

A: We need to use the above given two theorems to find the solution for the given congruence

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: 5) Which are true and which are false? a) 3XER,x^2-4x+5=0 b) VxɛZ,x^2 =0(mod4)orx^2 =1(mod4) c) Vx E…

A: In the given question we have to tell about some given statement which of them are true or which of…

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: