Question

Show that 9 n cannot end in 2 for any positive integer n.

Answer 1

Let, p(n) denotes the statement that 9ⁿ can not end with 2 for any positive integer n.
For n=1, p(1): 9¹=9, not ended with 2.
Let us assume that p(n) is true for n=k i.e., $9^{k}$ can not ended with 2.
For n=k+1, p(k+1):  $9^{k+1}$
=$9^{k}.9$
which can not be ended with 2 also since $9^{k}$ is not ended with 2.
Now p(1) is true and p(k+1) is true if p(k) is true. Then by the principle of mathematical induction 9ⁿ can not be ended with 2 for any positive integer n.