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 …
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 not divisible by 3 then n^2+2 is divisible by 3 Read More »
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. …
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 a power of 2 Read More »
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 …
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+1 is not Read More »
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 …
Prove that If n is a perfect square then n+2 is not Read More »
We will prove the given statement by direct method. Direct Proof First we will proof in (→) direction “If n is odd then 5n + 6 is odd” Next, we will proof in (←) direction “If 5n+6 is odd, then n is also odd” From the above 2 cases we can conclude that if n …
Prove that if n is a positive integer then n is odd if and only if 5n+6 is odd Read More »
Before proof, we will define some basic terminologies. Bipartite Graph: A graph whose vertices can be divided into 2 disjoint sets such that no two vertices in the same set are adjacent to each other is called Bipartite Graph. Proof We can prove the given statement by contradiction. Suppose you have a bipartite graph G. …
Prove that a graph which contains a triangle cannot be bipartite Read More »
In this article, we will write a program to calculate simple interest and compound interest. The formula for simple interest and compound interest is given below: First the program will ask the user to enter principal amount, time period and rate of interest. Then we will use the above formulas to calculate S.I and C.I.
