(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: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

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

A:

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

A:

Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: 37x = 101 (mod 151)

A:

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: How many divisors does (1000000! + 1) have that are less than 1000000?

A:

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

A:

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 or disprove 4|[(2²n-1 + 4)"]⁰

A:

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

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Q: 2x² – 3x = 9(mod13) X = (mod13)

A:

Q: Solve for X, show all calculation in binary number system only X = ((1110)₂-(0011)₂) (10011010)₂)…

A:

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

A:

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: x² + x = 13(mod17) X= * (mod17)

A:

Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's states that if p is a prime and a is a positive integer not divisible by p, than ap-1≡1 mod…

Q: The point x = 0 is a regular singular point of the differential equation. 1 x²y" + (²x + x²)x¹. y -y…

A:

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

A:

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

A:

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: Х1 + 2х2 + Зx3 — —7 2x1 – 3x2 – 5x3 = 9 — 6х1 — 8х2 + Хз 3— 22

A: We will find out the required solution of given system of equations.

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: How many consecutive zeroes are there at the end of 100! (100 factorial). For example, 12! -…

A: How many consecutive zeroes are there at the end of 100! We know that…

Q: 20) (x² – 3x – 28) ÷ (x + 5) 8 A) x-8+ x + 5 12 B) x-8 + x + 5 2 C) x – 8 – x + 5 11 D) x- 8+ - x +5

A:

Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

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: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

A:

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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Q: how: For all integers and , if n-m is even then  n^3-m^3  is even

A: We have given that for all integers n,m (n-m) is even.