Prove by mathematical induction that `n^(5)` and n have the same unit digit for any natural number n.

Explanation

We have to prove that `n^(5) - n` is divisible by 10.
For `n = 1, 1^(5) - 1 = 0` is divisible by 10.
Also, `n = 2 , 2^(5) - 2 = 30` is divisible by 10.
Thus, `P(1)` and `P(2)` are true.
`k^(5) - k = 10m"........"(1)`
Now, `(k + 1)^(5) - (k +1)`
`= k^(5) + 5k^(4) + 10k^(3) + 10 k^(2) + 5 k + 1 - k - 1`
`= (k^(5) - k) + 5 k(k^(3) + 1) + 10 k^(3) + 10 k^(2)`
`= 10 m + 10 k^(3) + 10 k^(2) + 5k(k^(3) + 1)` [Using (1)]
Clearly, `k(k^(3)+1)` is even for `k in N`.
Thus, `P(k+1)` is true whenever `P(k)` is true,
So, by the principle of mathematical induction, `P(n)` is true for any natural number n.