Question

If n is a natural number then 8^n - 3^n is divisible by which number?

Answer 1

Step 1: For n=1 we have 81−31=8−3=5 which is divisible by 5.

Step 2: Suppose (*) is true for some n=k≥1 that is 8k−3k is divisible by 5.

Step 3: Prove that (*) is true for n=k+1, that is 8k+1−3k+1 is divisible by 5. We have

8k+1−3k+1
=8∗8k−3∗3k

=5∗8k+3∗8k−3∗3k
=5∗8k+3(8k−3k)

Now we can say that 5∗8k is divisible by 5
​​​​​​From case 2 (8k-3k) is divisible by 5.
since it has the form 5∗p where p is an integer ≥1

And it is assumed that 8k−3k is divisible by 5, then 3(8k−3k) is divisible by 5 which means it has the form 5j

so we reduce the expression to

5p+5p=5(p+j)

which is of the form 5p, which is divisible by 5.

This is a step by step proof.
For checking which number is divisible we can use the trial and error method. Put n=1 then we get 5. Then put n = 2 we get 55 which also divisible by 5