Which of the following fractions has the decimal expansion completed?

Question

Which of the following fractions has the decimal expansion completed?

a) $1/(1024^1024)$
b) $1/(2222^2222)$
c) $1/(5555^5555)$
d) $1/(1500^1500)$

I know there should be power of 10 in the denominator, but I don't how to check if some of this fractions has it.

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Answer 1 · 2017-10-09T19:33:46

a) $1/(1024^1024)$

Note that $1024 = 2^10$

So:

$1/(1024^1024) = 1/((2^10)^1024) = 1/(2^10240) = 5^10240/10^10240$

which has a terminating decimal expansion with $10240$ decimal places.

All of the other options have factors other than $2$ or $5$ in the denominator.

Answer 2 · 2017-10-09T19:44:20

The correct answer is $A$ . See explanation.

A fraction can be converted to a decimal without a period if and only if the prime factorization of the denominator consists only of $2$ and $5$ .

In $B$ we have: $2222=2*11*101$ all raised to $2222$ ,

In $C$ we have $5555=5*11*101$ raised to $5555$

In $D$ we have $1500=2^2*3*5^5$ raised to $1500$

In $A$ the denominator can be written as $(2^10)^1024$ , so it is only the power of $2$