Question

Prove that there is no natural number for wch 4n ends with the digit zero

Answer 1

I'd the number 4n for any n were to end with the digit 0 it must divisible by 5. So the prime factorisation of 4n would contain the prime 5.But this is not possible as
4n=2n×2n
And the uniqueness of the fundamental theorem of algorithm guarantee that there is no prime other than 2 in the prime factorisation of 4n. Hence there is no natural number n for which 4n end with 0

Answer 2

Step-by-step explanation:

→ No, 4ⁿ can never end with the digit 0 for any natural number n .

→ If 4ⁿ ends with 0 then it must have 5 as a factor .

But, 4ⁿ = ( 2² )ⁿ = 2²ⁿ .

→ It shows that 2 is the only prime factor of 4ⁿ .

Also, we know from the fundamental theorem of airthematic that the prime factorisation of each number is unique .

So, 5 is not a factor of 4ⁿ .

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