Question

Show that 9^n cannot end with digit 0 for any natural number n

Answer 1

Let p(n) denotes the statement that 9ⁿ can not be end with digit 0 for any natural number n.
For n=0, p(0)=9⁰=1
For n=1, p(1)=9¹=9 which is not ended with 0.
Now, let us assume that p(n) is true for n=k i.e., $9^{k}$ can not be ended with 0.
For n=k+1,
$9^{k+1}$
=$9^{k}.9$
 which can not be ended with 0 since $9^{k}$ and 9 both are not ended with 0.
∴, p(k+1) is true alse.
Now, p(1) is true and p(k+1) is true if we assume that p(k) is true.
∴, By the principle of mathematical induction it can be said that 9ⁿ can not be ended with digit 0.