Question

Prove that there is no natural number for which 4^n ends with the digit 0

Answer 1

no,4n can't end with digit 0 because to end with 0 it must contain 2 and 5 as its prime factorisation but for 4 the prime factorisation is 2

Answer 2

Answer and explanation:

To prove : There is no natural number for which $4^n$ ends with the digit 0.

Proof :

We know that,

To ends with 0 the number must have 5 as a factor.

Now we can write number as,

$4^n=(2^2)^n$

$4^n=(2)^{2n}$

i.e. 2 is the only prime factor of $4^n$.

According to fundamental theorem of arithmetic that the prime factorization of each number is unique .

So, 5 is not a factor of $4^n$ .

Hence, $4^n$ can never end with the digit 0 .