If be rational number such that the prime factorization of be rational number such that the prime factorization of is not of the form is not of the form , where n, are non-negative integers. Then x has a decimal expansion which is terminating.

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True

False

Neither

Either

answer is B.

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Detailed Solution

Concept- We can solve the presented problem by employing a method of contradiction. In this case, if we take'

'as a prime factor of

and find a solution using this, and if we obtain a true response using this, then our answer for the actual problem will be false since we are doing it for contradiction.
A given fraction

is written as

.
Then from above we have:

In above we see that the denominator of the right hand side fraction term is a product of some power or 2 and 10 .
And we know that dividing any number with either by

, or we write as

result is always a terminating decimal expression.
Also we know that every number on division by either

or by

number ) result so obtain is always a terminating decimal expression.
Thus , we may conclude that the rational number with a prime element in the form of 2n times 5m is always a terminating rational number.
Hence, of the above assumption, we may conclude that if

lacks a prime component of the type 2n times 5m, it is always a non-terminating rational number.
Hence, the correct options is 2
.

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