How do you prove that N 3 5n is divisible by 6?

n^3+5n = n^3-n+6n = (n-1)n(n+1)+6n. Since (n-1)n(n+1) is the product of 3 consecutive integers, it is divisible by 6, so the sum is divisible by 6.

How do you prove math induction?

A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1.

How do you prove a number is divisible by 4?

The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4.

Which number divides n 3 5n for all n ∈ N?

Thus as the number n3+5n is always divisible by 2 & 3. thus it is always divisible by 6.

How do you show divisibility?

Divisibility Rules for some Selected Integers

1. Divisibility by 1: Every number is divisible by 1 1 1.
2. Divisibility by 2: The number should have 0 , 2 , 4 , 6 , 0, \ 2, \ 4, \ 6, 0, 2, 4, 6, or 8 8 8 as the units digit.
3. Divisibility by 3: The sum of digits of the number must be divisible by 3 3 3.

How do you prove by mathematical induction?

Steps to Prove by Mathematical Induction 1 Show the basis step is true. That is, the statement is true for n = 1 n=1 n = 1. 2 Assume the statement is true for n = k n=k n = k. This step is called the induction hypothesis. 3 Prove the statement is true for n = k + 1 n=k+1 n = k + 1. This step is called the induction step

What is the induction step for divisibility?

That is, the statement is true for n=1 n = 1. n=k n = k. This step is called the induction hypothesis. n=k+1 n = k + 1. This step is called the induction step b b? Since we are going to prove divisibility statements, we need to know when a number is divisible by another. So how do we know for sure if one divides the other?

Is 6 K + 4 divisible by 5?

In other words, we assume that6 k + 4 is divisible by 5 for some k.Next we let n = k+1. In this step we show that if it is true for n = k, then the result also holds for n = k+1.If n = k+1, we have that th LHS = 6 k+1 + 4 = 6 * 6 k + 4 = 6* (6 k + 4) – 20.

Is m – 1 = 4 divisible by 4?

Since m − 1 is divisible by 4 and 4 is divisible by 4, then this expression is divisible by 4. And there you are…