Fermat's theorem implies that for a prime p. 2P = 2( mod p). Will the converse be true that if 2P = 2( mod p) then p is prime? b. If f is a multiplicative function, then F(80) = f(8)f(10) Is the number 4926834923 the sum of two squares?

Q: (1) Determine the number Sm of solutions to the equation xyz = 1 (mod m) for each positive integer…

A: Given that:To find:The number  of solutions to the given equation for each positive integer…

Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: If 2n +1 is prime for some n ≥ 1, prove that n is a power of 2.(Primes of the form 2n +1 are called…

A: Given: 2n+1 is prime for some n≥1. To Prove: n is a power of 2.

Q: Let p be an odd prime. Prove that (a) If p = 1 (mod 5) or p = 4 (mod 5), then 5 is quadratic residue…

A: Given p is an odd prime. We have to prove (a). if p≡1(mod 5)  or  p≡4(mod 5), then 5 is a…

Q: Work the modular arithmetic problem. (9.7)(mod 5). (9.7)(mod 5) = (mod 5)

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Q: Prove that √2-1 1 1 √2-1 and hence the two rectangles are similar *Figure 1.18* Deduce that if a…

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Q: Prove that if x is a rational number andy is an irrational number, then x + y is an irrational…

A: As you have asked the problem ( b) only. I have solved this problem in two steps by showing the…

Q: (i)Find the inverse of 5(mod9)     (  ii)Hence or otherwise solve simaltaneous equation…

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Q: 4. Let p be an odd prime. (a) For any integer 1 <k <p-2, show that (+1) = (#¹) where k denotes the…

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Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: Find a d, he nce ged (2695,154) other wise express 26 95p +15479=d pigtz the in or

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Q: Work the modular arithmetic problem. (11.10)(mod 7). (11 10) (mod 7) = (mod 7)

A: (a)   (11.10)mod 7  =  110 mod 7                            The remainder when 110 is…

Q: - 1)/2 many integers: n, 2n, P=¹n. ∙n. 2

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Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

Q: (c) Does the polynomial x² - 27 = 0 (mod 1271) have any roots? Note that 1271 = 31 - * 41 is not a…

A: We use the Chinese Remainder Theorem to find the solutions to the congruence x2-27≡0(mod1271). Now…

Q: Randy wanted to change all the light fixtures in his home. He found a great deal at the local store…

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

Q: 5. Let f(x) = 2x² + x + 4. It is known that f(1) = 0(mod 7). (i). Find a solution to f(x) = 0(mod…

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Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: 4. Let the function g be given by Oniseven 9(n) n =1 (mod 4). n =3 (mod 4) 1 %3D 1

A: This is a question from Number Theory.

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: Let f(x) be a fixed integer polynomial, and let m be a fixed positive integer. Denote the number of…

A: This follows from basics of set theory. We give all the details in step 2.

Q: 2. Let f: Z-Z be given by the following rule: x+6 x=0 x = 1 x+8 x 2 f(x)=x+5 (mod 3) (mod 3) (mod 3)…

A: The given function is, fx=x+6x≡0 mod3x+5x≡1 mod3x+8x≡2 mod3 To Find: (a) Example to show that f is…

Q: For each part, use the data provided to find values of a and b satisfying a2 ≡ b2 (mod N), and then…

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Q: (b) Find the remainder of (10100) (10+00) mod 13.

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Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: How do I use the Chinese Remainder Thm to solve this system of congrunces below? x ≡ 1 (mod 3) x ≡ 4…

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Q: 2.3(i) Find the conjugates, moduli and arguments (to the nearest degree) of the following

A: 2.3(i)   (a). Given,                         3+5i.     To find the conjugate, modulus and argument…

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

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Q: Work the modular arithmetic problem. (8.9)(mod 4). (8.9)(mod 4) = (mod 4)

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Q: Exercise 6.3.4. Find all solutions modulo 10 to x² + 3x = 0 (mod 10). There are two distinct…

A: We employ the chinese remainder theorem to find the four solutions. Details typeset in latex in step…

Q: Which choice gives an example that supports the conjecture, and a counterexample that shows the…

A: The objective is to select the correct option from the given options. Given that, for any real…

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: Prove that 18000 + 28000 38000 Theorem (and not a calculator). 18000 + 58000 + +4° 8000 is a…

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Q: 18.) Compute fg (Note: each should only be one In 2₁, we write -1 as f(x)=x²-3x +2, term in the…

A: fg(x) = f(x)g(x)