Question

show that 9^n cannot end with for any natural n.

Answer 1

9 power n cannot end with o for an natural number n.if 9 power n ends with 0 ,it must be divisible by 5 and 2 i.e it must have prime factor as 2 and 5.but ,

9 power n =(3 squared) power n

The only prime factor is 3 .

By fundamental theorem of arithmetic the promise factorisation of each number is unique so 9 power n cannot end with 0

Answer 2

9^n can be written as (3*3)^n.
But due to fundamental theorem of arithmetic...it promises that factors are unique.And There MUST be both 2and5 as factors...only then a number can end with 0