Question

Prove by mathematical induction for all positive integral values of x, x(x^2+5) is divisible by 6.

Answer 1

the first positive integer is 1
so have you proved it for x=1 yet?

Answer 2

@ganeshie8
@ParthKohli

Answer 3

No I actually am not aware of mathematical induction techniques.

Answer 4

You prove it for the base case.
Then you assume it is true for some k within the chosen domain of the problem.
Then you show it is true for the case x=k+1.

Answer 5

the base case being x=1 in this case since it is the first positive integer

Answer 6

Can you explain it in a little more detail

Answer 7

I know you want it proven by induction, but here is a quick way to convicne yourself that the given statement is indeed true :
\[x(x^2+5)\equiv x(x^2-1) = (x-1)x(x+1)\]

Recall that the product of any 3 consecutive integers is divisible by 3 factorial

Answer 8

How is +5 equals to -1

Answer 9

5 = 6 - 1

since 6 is dividible by 6, you can replace it by 0

Answer 10

Perhaps this is more clear :
\[x(x^2+5) = x(x^2-1+6) =x(x^2-1) + 6x = (x-1)x(x+1) + 6x \]

Again, this has nothing to do with induction...

Answer 11

@ganeshie8 Okay now can you explain it woth induction

Answer 12

You need to know how/why induction works first

Answer 13

Yeah

Answer 14

Let me ask you a question

Answer 15

Consider the set of negative integers.

Is there a minimum element in this set ?
If a minimum element exists, what is it ?

Answer 16

Yes and I suppose 1, and what's the name of the technique you used above to prove the statement

Answer 17

do you mean, 1 is the minimum element in the set of "positive integers"

Answer 18

Oh they are saying that you have to use positive integral values

Answer 19

and what's the name of the technique you used above to prove the statement

Answer 20

Earlier proof is based on simple divisibility arguments

We expressed the given expression as a product of 3 consecutive integers; then we used the fact that product of any 3 consecutive integers is divisible by 3 factorial.

Answer 21

don't pay too much attention to it
you want to do this using induction right ?

Answer 22

Yes

Answer 23

Lets get back to induction

Answer 24

sure

Answer 25

Suppose I give you a machine that does this :

if you input a positive integer \(k\), the machine spits out the next integer \(k+1\)