Question

What can you say about the prime factorization of the denominator of $27.\overline{142857}$?

Answer 1

Hint: We have a repeating decimal number. We will first convert this number into a fraction. There are ways to convert different types of decimal numbers into fractions. We will be using the process that involves converting repeating decimals into fractions. Then we will look at the prime factorization of the denominator.

Complete step by step answer:
To convert the given repeating decimal number, we will write the given number in the following manner, $27.\overline{142857}=27+0.\overline{142857}$.
So, the numerator of the fraction will have the number 142857. Since all the six digits after the decimal are repeating, the denominator of the fraction will have six 9's. As there are no non-repeating digits after the decimal, the 9's in the denominator will not be followed by any zeros.
So, we have $0.\overline{142857}=\dfrac{142857}{999999}$.
Therefore, the given decimal number will be converted into fraction as follows,
$27.\overline{142857}=27+0.\overline{142857}$
We will substitute the fraction value of the repeating decimal. So, we now have the following,
$27.\overline{142857}=27+\dfrac{142857}{999999}$
Taking the LCM of the above expression, we get
$27.\overline{142857}=\dfrac{26999973+142857}{999999}$
Simplifying this expression, we have
$27.\overline{142857}=\dfrac{27142830}{999999}$
Next, we have to look at the prime factorization of the denominator. The denominator is 999999. This number can be factorized by prime numbers as follows,
$999999=3\times 3\times 3\times 7\times 11\times 13\times 37$.

Note: It is crucial to follow the correct method while converting a decimal number into a fraction. The calculations involve large numbers. Therefore, it is useful to write the calculations explicitly so that minor mistakes can be avoided. While factoring any number into its prime factors, it is useful to know the divisibility tests.