Prove that for every n ∈ N: a) (10^n + 3 * 4^(n+2)) ≡ 4 mod...

Prove that for every n ∈ N:

a) (10^n + 3 * 4^(n+2)) ≡ 4 mod 19, [note that 4^3 ≡ 1 mod 9]

b) 24 | (2*7^(n) + 3*5^(n) - 5),

c) 14 | (3^(4n+2) + 5^(2n+1) [Note that 3^(4n+2) + 5^(2n+1) = 9^(2n)*9 + 5^(2n)*5 ≡ (-5)^(2n) * 9 + 5^(2n) *5 ≡ 0 mod 14]

Expert Solution

Colby Messinger answered 1 year ago

Prove that 3^n + 7^(n−1) ≡ 4 (mod 12) for all n ∈ N+.

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod...

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod 4) or n ≡ 3 (mod 4), then n is not a perfect square.

1. compute the least non-negative residue of 4^n (mod 9) for n=1,2,3,4,5.... prove that 6*(4^n)=6 (mod...

1. compute the least non-negative residue of 4^n (mod 9) for n=1,2,3,4,5.... prove that 6*(4^n)=6 (mod 9) for every n>0. 2. find nice tests for divisibility of numbers in base 34 by each of 2,3,5,7,11,and 17. 3. in Z/15Z, find all solutions of : (i) [36]X=[78]. (ii) [42]X=[57] (iii) [25]X=[36] 4. in Z/26Z, find the inverse of [9], [11], [17], and [22] 4. write the set of solutions of x=5 mod24. x=17 (mod 18) for all equation line, there are...

Prove or disprove that 3|(n 3 − n) for every positive integer n.

Prove: If a1 = b1 mod n and a2 = b2 mod n then (1) a1...

Prove: If a1 = b1 mod n and a2 = b2 mod n then (1) a1 + a2 = b1 + b2 mod n, (2) a1 − a2 = b1 − b2 mod n, and (3) a1a2 = b1b2 mod n.

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

Prove true or false. For each natural number n, ((n5/5)+(n^4/2)+(n^3/3)-(n/30)) is an integer

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 )....

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 ). Your answer must clearly specify the constants c and n0. b) Let g(n) = 27n 2 + 18n and let f(n) = 0.5n 2 − 100. Find positive constants n0, c1 and c2 such that c1f(n) ≤ g(n) ≤ c2f(n) for all n ≥ n0. Be sure to explain how you arrived at the constants.

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for...

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for all integers n Show all work

Determine the solution of the following equation mod N. 1.7x≡2 mod 15, where N= 15 2.x≡8...

Determine the solution of the following equation mod N. 1.7x≡2 mod 15, where N= 15 2.x≡8 mod 11, x≡3 mod 19, where N= 209 3.x≡2 mod 7, x≡2 mod 11, x≡1 mod 13, where= 1001

Question