Prove that 3 divides n^3 −n for all n ≥ 1.

Expert Solution

Colby Messinger answered 2 years ago

Prove That For All Natural Numbers A > 1 And B > 1, If A Divides...

Prove That For All Natural Numbers A > 1 And B > 1, If A Divides B Then A Does Not Divide B+1 (prove by contradiction)

Use a direct proof to prove that 6 divides (n^3)-n whenever n is a non-negative integer.

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

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n...

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n + 3⋅9^n.

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

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a...

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a positive integer. HINT: 25 ≡ 4(mod 21)

Prove by contraposition and again by contradiction: For all integers a,b, and c, if a divides...

Prove by contraposition and again by contradiction: For all integers a,b, and c, if a divides b and a does not divide c then a does not divide b + c Elaboration with definitions / properties used would be appreciated! Thanks in advance!!

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 that for all integers n ≥ 2, the number p(n) − p(n − 1) is...

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is equal to the number of partitions of n in which the two largest parts are equal.

prove 2 is a factor of (n+1)(n+2) for all positive integers

Question