Question

Whether \[8\] power \[n\], where \[n\] is a natural number, can end with zero ?

Answer 1

Hint: In this question we need to find whether \[{8^n}\] can end with zero, that is, its unit digit can be \[0\] or not. We can find this by using the divisibility test. The divisibility rule of \[10\] states that a number is divisible by \[10\] only when its unit digit is \[0\]. Thus if \[{8^n}\] is divisible by \[10\] then it can end with \[0\].

Complete step by step answer:
We are given \[{8^n}\], where \[n\] the exponential. Now find the factors. The factors of \[10 = 2,5\], these are prime numbers too. Thus if \[{8^n}\] is divisible by 10, it has to be divisible by \[2\] and \[5\] both. The factors of \[8 = 1,2,4,8\]. Here we can find only \[2\] as a factor, it means that \[8\] is only divided by \[2\] and not by \[5\] therefore it cannot be divisible by \[10\] too.
\[8 = 2 \times 4\]
The prime factorization of \[8 = 2 \times 2 \times 2\].
The prime factorization of \[{8^n} = {(2 \times 2 \times 2)^n}\].
By this observation, we can say that \[{(2 \times 2 \times 2)^n}\] is divisible by \[2\] and not by \[5\]. Therefore \[{8^n}\] is not divisible by \[10\].

Hence \[{8^n}\] cannot end with zero.

Note:Any number to have its unit place be zero, then it must be multiplied by \[10,100,1000\] and so on. The prime factors of all these numbers are always \[2\] and \[5\].If a number is the factor of both \[2\] and \[5\] then its unit place can be \[0\] and if only one among them is present as a factor its unit place cannot be \[0\]. A number is said to be a prime number if the number exactly has two factors, \[1\] and the number itself.
Factors of \[2 = 1,2\].
Factors of \[5 = 1,5\]. Thus \[2\] and \[5\] are prime numbers.
We found that \[{8^n}\] cannot end with zero for any natural number \[n\]. Counting numbers are called natural numbers. Thus \[1,2,3,....\] are called natural numbers.
\[{8^1} = 8\]
\[\Rightarrow {8^2} = 64\]
\[\Rightarrow {8^3} = 512\]
\[\Rightarrow {8^4} = 4096\] and so on.
Thus any value for \[n\], \[{8^n}\] units cannot be zero.