# Discrete mathematics

## Prove that n^3+2 is not divisible by 8

Given n3+ 2 is not divisible by 8 ∀ n ∈ N Proof For value of n, we have 2 cases CASE 1 : n is odd If n is odd then n3 is also odd. We know ODD + EVEN = ODD. Therefore, n3+2 is odd. 8 cannot divide an odd number. We can …

## Prove by induction n^3+(n+1)^3+(n+2)^3 is divisible by 9

Given n3 + (n + 1)3 + (n + 2)3 is divisible by 9 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 13 + (1 + 1)3 + (1 + 2)3= 13 + 23 + 33 = 1 + 8 + 27 = 36 36 …

## 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 …