(7) Let m≥ 1 and let R= {0, 1, 2,...,m - 1}. (a) Prove that if r1, T2 € R and r₁ r2 (mod m), then r₁ = 12. for

Q: coprime

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

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Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: Prove that AB #BA for [2 4 3 2] A = 1 1 0 3 14 6 2 5 B 1 2 0 315 3 2 4 L1 1 6

A: We have given two matrices A and B to prove that AB is not equal to BA.Matrixes are given below :

Q: prove for all ne J, 1+ 3 + 5 +..+(2n-1) = n² by induction

A: The statement is for all To prove :The above statement is true using mathematical induction.

Q: 13. Use induction on k to EIR", then X₁1 | x,t... Xklls x + prove that if +||XK||

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Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

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Q: x and y are two positive integers such that y > 5x. Select the expressions that are not equal to…

A: We choose x and y such that we are able to discard options.

Q: Prove that for all n € N, 91(4" + 6n + 8) and n € N, 31(n³-n).

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Q: Consider certain real numbers ui and vi, i = 1,..., n, n <∞0. Show that (u₁v₁ + U₂v₂)² < (u² + u²)…

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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: Let m 2q be an even positive integer. Prove that every integer is congruent modulo m to a unique…

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Q: For allne z, gcdin,nt)=l.

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Q: verify that det(A B) = det(B A) and determine whether the equality det(A + B) = det(A) + det(B)…

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Q: Let A = 1 = {n+ (−1)n 2" | REN}. n Compute sup A and inf A if they exist.

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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: prove f(x)=2x (mod 7) is injective

A: To prove that the function f(x) = 2x (mod 7) is injective, we need to show that for any two distinct…

Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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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: 1 S = 0 1000 100,000 10 € = (†)* (k ≥ 1). " 2n-1 (1) ²¹—¹, ...}, 10

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Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: (7) Let m≥ 1 and let R= {0, 1, 2,...,m - 1}. (b) Prove that for every integer a there is a unique…

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Q: (14) Let E denote the set of even integers, O denote the set of odd integers, and x, y € Z be any…

A: I have given the answer in the next step. Hope you understand that