Let n be a positive whole number. Which of the following...

Let $n$ be a positive whole number. Which of the following conditions imply that the $n$ is divisible by $6 ?$ (a) $n$ is divisible by 3 (b) $n$ is divisible by 9 (c) $n$ is divisible by 12 (d) $n^{2}$ is civisible by 12 (e) $n=24$ (f) $n$ is even and divisible by 3 (g) $n=m^{3}-m$ for some positive whole number $m$ Which of $(a)-(g)$ are logically equivalent to the statement ' $n$ is divisible by $6^{\prime} ?$

Let $n$ be a positive whole number. Which of the following conditions imply that the $n$ is divisible by $6 ?$
(a) $n$ is divisible by 3
(b) $n$ is divisible by 9
(c) $n$ is divisible by 12
(d) $n^{2}$ is civisible by 12
(e) $n=24$
(f) $n$ is even and divisible by 3
(g) $n=m^{3}-m$ for some positive whole number $m$
Which of $(a)-(g)$ are logically equivalent to the statement ' $n$ is divisible by $6^{\prime} ?$

Key Concepts

Divisibility

This key concept refers to one integer being a factor of another, meaning the division results in an integer with no remainder. An understanding of divisibility is essential when determining the conditions under which a given number can be divided evenly by another, such as in the case of testing divisibility by 6.

Divisibility Rules for Composite Numbers

This concept involves the idea that to check divisibility by a composite number like 6, one can instead check divisibility by its prime factors—in this case, 2 and 3. Recognizing how composite divisibility breaks down into simpler tests is crucial for analyzing and proving statements about divisibility.

Even Numbers and Divisibility by 2

Being even means that a number is divisible by 2. This is a fundamental property used in many divisibility tests, including those for composite numbers, because many composite divisibility rules require that the number satisfy a condition related to evenness.

Logical Equivalence

Logical equivalence refers to a relationship between two statements where each one implies the other. In the context of divisibility problems, determining which conditions are logically equivalent to a number being divisible by 6 means identifying conditions that are both necessary and sufficient for the divisibility by 6.

Necessary and Sufficient Conditions

This concept distinguishes between conditions that must be met for a property to hold (necessary) and those that guarantee the property if they are met (sufficient). In the analysis of divisibility, it is important to identify whether a given condition both necessitates and ensures divisibility by 6, thereby fully characterizing the property.

Transcript

00:01 We want to show that n -cube minus n is divisible by six for all n in the positive integers.

00:06 So we want to check the base case, which is n equals 1, since we're talking about the positive integers, then 1 -1 -1 -is equal to 0, which is divisible by 6.

00:21 So n -equals 1 is true.

00:25 We assume this is true up to n -equals k.

00:32 That would tell us k cubed minus k is divisible by six.

00:42 Then for n equals k plus 1, that would mean k plus 1 cubed minus k plus 1.

00:54 And we want to try to show that this thing is also divisible by 6.

00:59 We'll have to expand it.

01:01 So i'm going to take out a common factor of k plus 1.

01:04 And left with k plus 1 squared minus 1.

01:10 And that works out to k plus 1 times k squared plus 2k.

01:17 So this is easy to distribute.

01:19 This is k cubed plus 3k squared plus 2k.

01:30 And we want to find a way to use that inductive step.