Question

Which of the following fractions has the decimal expansion completed?

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.

Answer 1

a) `1/(1024^1024)`

Explanation:

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

The correct answer is `A`. See explanation.

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`