Question

check whether 6^n can end with the digit 0 for any natural number n

Answer 1

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Answer 2

$\bf\huge{\underline{\underline{Question}}}$

Check whether 6^n can end with the digit 0 for any natural number n.

$\bf\huge{\underline{\underline{Solution}}}$

Here, n is a natural number and let ${6^n}$ ends with digit 0

∴ ${6^n}$ is divisible by 5.

But the prime factors of 6 are 2 and 3. i.e., 6 = 2 × 3

=> ${6^n}$ = ${(2 × 3)^n}$

i.e., In the prime factorisation of ${6^n}$, there is no factor 5.

So, by the fundamental theorem of Arithmetic, every composite number can be expressed as a product of primes and this factorisation is unique apart from the order in which the prime factorisation occurs.

∴ Our assumption that ${6^n}$ ends with digit 0, is wrong.

Thus, there does not exist any natural number n of which ${6^n}$ ends with zero