Question

Look at several examples of rational numbers in the form \[\dfrac{p}{q},q \ne 0\], where \[p\] and \[q\] are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property \[q\] must satisfy?

Answer 1

Hint: We will first consider the given statement in the question that is the rational numbers of the form \[\dfrac{p}{q},q \ne 0\], where \[p\] and \[q\] are integers with no common factors other than 1 and having terminating decimal representations. As we have to find the property that \[q\] must satisfy, we know that there is no common factor other than 1 which means there is no other number which will divide both \[p\] and \[q\] and also maintain the decimal representation implies that prime factorization of \[q\] has only powers of 2 or power of 5 or both and this will give us the required property that \[q\] will satisfy.

Complete step-by-step answer:
We will first consider the given statement that is the rational numbers of the form \[\dfrac{p}{q},q \ne 0\], where \[p\] and \[q\] are integers with no common factors other than 1 implies that there will no common divisor between \[p\] and \[q\] other than 1 which will divide both the integers.
Also, it is given that the rational numbers in the form \[\dfrac{p}{q},q \ne 0\] have the terminating decimal expansion which means that the number will terminate after some time and there is no pattern which will be repeated.
We need to find the property where \[q\] must satisfy the above statements.
Thus, we get that the property that \[q\] must satisfy in order that the rational numbers of the form \[\dfrac{p}{q},q \ne 0\], with no common factor other than 1, have maintaining decimal representation is prime factorization of \[q\] has only powers of 2 or power of 5 or both.
That is, we get,
\[{2^m} \times {5^n}\] where \[m = 1,2,3...\] or \[n = 1,2,3...\].
Here, we can consider one example such as \[\dfrac{1}{4},\dfrac{{37}}{{25}},\dfrac{{17}}{{20}}\] which has terminating decimal expansion.
Thus, we get,
\[\dfrac{1}{4} = \dfrac{1}{{2 \times 2 \times 1}}\] , \[\dfrac{{37}}{{25}} = \dfrac{{37 \times 1}}{{5 \times 5 \times 1}}\] and \[\dfrac{{17}}{{20}} = \dfrac{{17 \times 1}}{{2 \times 2 \times 5 \times 1}}\]
Hence, we can see that the denominator is either power of 2 or power of 5 or both.
Thus, we can conclude that in the rational number of the form \[\dfrac{p}{q}\], \[q\] is either power of 2 or power of 5 or both.


Note: Considering an example in such questions makes the concept clearer. We have to consider only those integers of the form of rational numbers which are terminating that is no pattern is followed or repeated after dividing the number gets terminated.