Question

if x = P /q b are rational number whose decimal expansion terminate then the prime factorization of q is of the form a)2m×3n b)2m×5n c) 3m×5n d)5m×7n​

Answer 1

Answer:

if we have a rational number of the form p/q , and the prime factorisation of q is 2^m.q^n , where m and n are non negative integers, then p/q have a terminating decimal expansion.

as we can easily agree with the fact that a rational number of the form a/b , where b is a power of 10 will have a terminating decimal expansion.

so it seems to make sense to convert a rational number of the form p/q , where q is of the form 2^m.q^n to an equivalent rational number of the form a/b , where b is a power of 10.

now let us consider following examples:

38=323=3*5323*53=3*5310313125=1353=13*2323*53=13*23103780=710*23=7*5310*23*53=7*5310*103=7*53104

therefore we can see that how we can convert a rational number of the form p/q , where q is of the form 2^m*5^n to a rational number a/b , where b is a power of 10.

therefore the decimal expansion of such a rational number terminates.

hope this helps you

Answer 2

Therefore the prime factorization of 'q' should be of the form $2^m *5^n$. ( Option-b )

Given:

A rational number x = p / q  whose decimal expansion terminate

To Find:

The representation of 'q'.

Solution:

The given question can be answered very easily as shown below.

Given rational number x = p / q  whose decimal expansion terminate.

We don't know anything about 'p', it can be odd or even.

To get the non-recurring or terminating decimal values in the number 'q' should be prime factorized as $2^m *5^n$.

Because when a number is divided by '2'm, decimals do not recur at any cost.

Similarly, if any number is divided by '5', the decimals in the number do not recur.

So the multiplication of both these primes also gives a rational number with nonrecurring decimals.

Therefore the prime factorization of 'q' should be of the form $2^m *5^n$.

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