(5) (Extra credit, 5 points) Show that 102n-1 +1 is divisible by 11 (namely, 102n−1 +1 = 11 ·an, where a₁ = 1 and an+1 = 100an 9 for all natural numbers n = 1, 2, ...). [Hint: Show that 102n+1+1=100 (102n−1 + 1) - 99.]

Q: how that in the domain of integers of the form a+bsqrt(-17) the factorization…

A: The  objective is to show that 13 is irreducible. The norm of 13 is 13=132=169 So if 13 is reducible…

Q: [1+6n A" 7 4 4n -9 -5 -9n 1-6n | If , show that for every n natural number.

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Q: 9(x + 6) - 41 = 75 Prove x - 62/9)

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Q: For a natural number n, we define n! = 1-2-3 (n − 1). n. - For example: 2! = 1-2, 3! = 1-2-3 = 6,…

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Q: prove

A: Here we use, 3+3=9 and 6+3=9 We have to prove,3×3=9 by peano's axiom for natural number.

Q: 37x = 101 (mod 151)

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Q: Prove (kindly provide reason/s) that for any real numbers m and n, (-m ) + ( - n ) = - ( m + n)

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Q: Maximise Z = 5x1 + 6x2 Subject to X1 + x2 0 and integer

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Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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Q: Prove by induction that 32n 1 is divisible by 8 for Vn 20

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Q: If n is a positive integer prove that 3^(2n) - 26n -1 is divisible by 676

A: if n is a postive integer prove that 3^(2n) - 26n -1 is divisible  by 676

Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: Prove by cases For any integer n, n^2 + 5 is not divisible

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Q: Let k be an positive integer. Prove that k(7k^2 + 5) + 6 is divisible by 6

A: Let k be an positive integer. Prove that k(7k^2 + 5) + 6 is divisible by 6

Q: Prove that √1+i√3+√1-i√3 = √6

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Q: Prove or disprove 4|[(2²n-1 + 4)"]⁰

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Q: For 7-10 Given a_1= 3, a_2= 7 and a_n= [2a] _(n-1) + a_(n-2) for n 23, 7) Find a_3 =,* Your answer…

A: First Question: (7) Given a1 = 3 , a2 = 7 and  an = 2an-1+an-2 then a3 = 2a2+a1 = 2*7+3 = 17 Hence…

Q: Prove or disprove: There exists an integer a such that 14 | (3a – 5) and 21 | (2a +3).

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Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

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Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: Let n be a positive integer. Using congruences, show that 3n+3 +23n+3 is divisible by 7.

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Q: Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4…

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Q: Which of the following gives the Binet form of the nth Fibonacci number? O Fn = Fn = Fn = Fn =…

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Q: S(x + Vx)dx 73 6.

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Q: Show that the weighted arithmetic mean of first 'n' natural numbers whose weight are equal to the…

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Q: If a and b are distinct integers prove that a-b is a factor of an -b" whenever n is positive integer

A: if a and b are distinct integers prove that a-b is a factor of a^n - b^n whenever n is positive…

Q: One of the striking properties of Fibonacci numbers is that gcd(Fm, Fn) = Fd, (*) where d = gcd (m,…

A: 1. We see,        F9 = 34    [Fibonacci series : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,. . . . . ]…

Q: 5. Prove that n² is an even number if and only if 3n + 1 is an odd number by proving a) If n² is…

A: To Prove: We have to prove that n2 is an even number if and only if 3n+1 is an odd number.

Q: While proving (a*2(2a+1 is an integer for the case a = 6q +1 which of the following will be obtained…

A: We are given with a(a+1)(2a+1)6 We need to substitute a=6q+1

Q: In x2 10) + In (9) = In (10)

A: Definition used-             ln(x)+ln(y)=ln(xy)

Q: 5 =5 Hence, 6"-1 is divisible by 5 is (10.). for n=1 Part 2: Assume that 6" - 1 is divisible by 5…

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Q: x+(x+1)+(x+2)+(x+3)+(x+4)=5x+10 What does this prove about the sum of these natural numbers?

A: The sum of the given equation is calculated as follows.

Q: If [5 - -8 A = [611] 0 " then A- 1 =

A: The objective of the question is to find the value of 'A - 1' given the equation '5 - -8A = 611'.

Q: Please provide justified answer asap:::: Show that n is odd if and only if n^2+4n+1 is even.

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