Question

show that 9^ n cannot end with digit zero for any natural number n

Answer 1

9n can never end with the digit 0 for any natural number n because for a number to end with zero it should be divisible by 10 and therefore its factors should be 5 and 2.

But as 9 has only 3 as its factor, increasing its power will not make 5 as its factor.

Thus it can never end with 0.

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Answer 2

For a number to end with digit zero the prime factorisation of the number must contain the factor 2^n 5^n but the prime factorisation of 9^n is 3^2n . by the fundamental theorem of arithimetic the nos 9^n can never end with digit zero .

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