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

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

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

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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