## Prove by induction that 6^n-1 is divisible by 5

Given 6n – 1 is divisible by 5 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 6n – 1 = 61 – 1 = 5 5 is divisible by 5. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove n^2+2 is not divisible by 4

Given n2 – 2 is not divisible by 4 ∀ n ∈ N Proof For value of n, we have 2 choice CASE 1 : n is even If n is even, then we can write n as n = 2x, x ∈ N n2 – 2 = 4×2 – 2 = 2( 2×2 – …

## Prove 4^n-1 is divisible by 3

Given 4n – 1 is divisible by 3 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 4n – 1 = 41 – 1 = 3 3 is divisible by 3. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove that n^3-n is divisible by 6

Given n3 – n is divisible by 6 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 n3 – n = 13 – 1 = 0 0 is divisible by 6. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove that if n is not divisible by 3 then n^2+2 is divisible by 3

For the value of n, we have 3 choices Case 1 : n % 3 = 1 Let, P = (n2+2) % 3 P = ( ( n2 ) % 3 + 2 % 3 ) % 3 P = ( ( n % 3 ) ( n % 3 ) + 2 ) % …

## Prove that if n is odd then 8 divides n^2-1

Given if n is odd then 8 divides n2-1 Proof We can proof the given statement directly. Let n ∈ O Let, P = n2 – 1 We need to prove that P is divisible by 8 P = n2 – 12 P = (n + 1)(n – 1) We know that n is odd. …

## Prove that if 2^n+1 is prime then n is a power of 2

We will prove the given statement by contradiction. Let 2n + 1 be a prime number. Let it be P P = 2n + 1 Assume n not be in power of 2. That means one of the multiple of n must be an odd number ( other than one ). Therefore, we can write …

## Prove that if 2^n-1 is prime, then n is prime

We will prove the given statement by contradiction. Suppose 2n – 1 to be prime. Let it be P P = 2n – 1 Assume n is not a prime number. If n is not a prime numer then we can write n = x*y | x, y > 0. From equation 3, it is …

## Prove that If n is a perfect square then n+1 is not

In this article, we will prove that if n is a perfect square then n + 1 is not a perfect square for n > 0. If n is a perfect square, then there exists an integer x such that x2 = n … (1) Now, assume n + 1 to be a perfect square …

## Prove that If n is a perfect square then n+2 is not

In this article, we will prove that if n is a perfect square then n + 2 is not a perfect square for n >= 0. If n is a perfect square, then there exists an integer x such that x2 = n … (1) Now, assume n + 2 to be a perfect square …