13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

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Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Question 6: Use the Laplace transform to solve the following second-order differential equation:…

A: The given differential equation is:d2vtdt2+2dvtdt+10vt=3cos2t v0=1 and dv0dt=-2. We have to solve…

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A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

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Q: 14. Find all 2<x<19999 such that x=-x(mod 20000) Ans:

A: To find all x where 2≤x≤19999 such that x2≡-xmod 20000.

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A: By the help of even and odd integer's we prove the result. The answer is given as follows :

Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: 5. Find an integer x between 0 and 776 such that x = 13 (mod 21) x = 5 (mod 37)

A: Set N = 21 × 37 = 777. Then , m1  = N/21 = 777/21 = 37 and m2 = N/37 = 777/37 = 21 Now we need to…

Q: 4b. Show that there exists i and j with i < į such that the consecutive sum p;+pi+1++p; is divisible…

A: We are given the following. There are 51 software students taking MATH 211.  Let p1, p2, ... , p51…

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Q: Prove that 6 divides n³ - n whenever n is a nonnegative integer.

A: In this problem we have to prove that  divides  where  is a nonnegative integer.

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: 11. Prove that a¹ = 1 (mod 5) for every integer a that is not divisible by 5.

A: We have to prove  for every integer  that is not divisible by

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Q: 6.25. every integer n> 4. - 9n - 10) for every nonnegative integer n.l Prove that 1++++ <2- for…

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Q: 1. Compute the least nonnegative residue of 10k + 1 modulo 13 for k 1, 2, 3, 4.

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Q: 15. Give examples of the following. Justify your answers.: (b) A set X such that |P(X)| = 8. (c)…

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Q: 8. If a and b are odd integers, then 8|(a² – b²) -

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Q: How many positive integers n less than or equal to 11,347 satisfy d(n) 1 (mod 2)?

A: We need to find a number of positive integers n≤11347 such that d(n)≡1(mod 2) i.e., , k∈Z.So, we…

Q: mod 1537). 2. Frove the following statements: a) For any integer a, the units digit of a? is 0, 1,…

A: We can solve this expression of  numerical analysis

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: [n' 2 32. For every integer n, SIN 3

A: Introduction: A ceiling function is one that returns the smallest successive integer. In other…

Q: 6. For any positive integer n > 1 one has: (2n)! (n!)² n+1 <

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Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: A) Express the following using favtorials and simplify: ([n] [2]). B) Show the following using…

A: Consider the combination n r as nCr. Note that, nCr=n!r!×n-r!. A) n…

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

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Q: If x – 1 is an even integer, which of the following numbers must be odd? I. x(x - 1) II. 3(x - 1)…

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Q: Find the integer n that has the following properties: i. gcd(n, 10) = 1 ii. iii. 9 = n38 (mod 10) -9…

A: Concept: The smallest positive integer d that can be a solution or that can be represented as a…

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

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Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: - Prove that for every positive integer 11, 1.2.3+2.3.4+... +n(n+1) (n + 2) = n(n+1)(n+ 2)(n+3)/4.

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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)

Q: 5 is a quadratic non- residue mod 17

A: The quadratic residues mod17 are { 1,2,4,8,9,13,15,16}

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A: We use the fact that if gcd(a,p)=1 then a^{p-1}= 1 (mod p)