00:01
N to the 5 minus n is divisible by 5 for all positive integers n.
00:05
So we check the base case.
00:07
That would be n equals 1.
00:09
So on the left, really, it's 1 to the 5 minus 1, which is 0.
00:14
That is divisible by 5 because 0 is divisible by anything.
00:20
So we know that n equals 1 is true.
00:25
We assume true up to to n equals k.
00:34
So then we can say k to the 5 minus k is divisible by then n equals k plus 1.
00:54
That would be written as k plus 1 to the power of 5 minus k plus 1.
01:01
And we've got to expand this so we can take out a common factor of k plus 1 to make our jobs a little bit easier...
