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

Expert Solution

Colby Messinger answered 1 year ago

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]

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: 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 3 divides n^3 −n for all n ≥ 1.

Prove by induction that 14^n + 12^n −5^n is divisible by 7 for all n >0

For m, n in Z, define m ~ n if m (mod 7) = n (mod...

For m, n in Z, define m ~ n if m (mod 7) = n (mod 7). a. Show that -341 ~ 3194; that is to say 341 is related to 3194 under (mod 7) operation. b. How many equivalence classes of Z are there under the relation ~? c. Pick any class of part (b) and list its first 4 elements. d. What is the pairwise intersection of the classes of part (b)? e. What is the union of...

0 mod 35 = 〈0 mod 5, 0 mod 7〉 12 mod 35 = 〈2 mod...

0 mod 35 = 〈0 mod 5, 0 mod 7〉 12 mod 35 = 〈2 mod 5, 5 mod 7〉 24 mod 35 = 〈4 mod 5, 3 mod 7〉 1 mod 35 = 〈1 mod 5, 1 mod 7〉 13 mod 35 = 〈3 mod 5, 6 mod 7〉 25 mod 35 = 〈0 mod 5, 4 mod 7〉 2 mod 35 = 〈2 mod 5, 2 mod 7〉 14 mod 35 = 〈4 mod 5, 0 mod 7〉...

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.

Use induction to prove that 8^n - 3^n is divisible by 5 for all integers n>=1.

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2) Prove that a finite set with n elements has 2n subsets (3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

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