Question

If these numbers form positive odd integer 6q+1, or 6q+3 or 6q+5 for some q, then q belongs to:
(a) Integers
(b) Rational
(c) Real
(d) None of these.

Answer 1

Hint: Try to recall the definition of rational, integers and real numbers. Use the property that the sum or difference of a rational and an irrational number is always an irrational number. Also, if we multiply a rational with an irrational number the result is irrational.

Complete step-by-step answer:
Before moving to the options, let us talk about the definitions of rational numbers followed by irrational numbers.
So, rational numbers are those real numbers that can be written in the form of $\dfrac{p}{q}$ such that both p and q are integers and $q0$ . In other words, we can say that the numbers which are either terminating or recurring when converted to decimal form are called rational numbers. All the integers fall under this category.
Now, moving to irrational numbers.
Those real numbers which are non-terminating and non-recurring are termed as irrational numbers.
The roots of the numbers which are not perfect squares fall under the category of irrational numbers. $\pi \text{ and }e$ are also the standard examples of irrational numbers.
Now moving to the solution.
It is given that 6q+1, or 6q+3 or 6q+5 must be a positive odd integer and we know that if we add or subtract an integer from another integer, the result is an integer. So, 6q must be an integer.
Now, for 6q to be an integer, we know that q cannot be an irrational number, as if we multiply a rational with an irrational number the result is irrational. So, q must be rational. However, it is not necessary that q must be an integer, it can be a rational number (fraction) such that 6q is an integer and is even too, for example: $q=\dfrac{1}{3}$ . As when 6q will be even then only the numbers 6q+1, 6q+3 and 6q+5 can be odd, as odd added with even given an odd number as a result.
Therefore, we can conclude that q belongs to the set of rational numbers for sure. As it belongs to rational numbers, it will belong to real numbers as well, because rational numbers are a subset of real numbers.
Hence, the answer is option (b) and (c).

Note: It is important that you understand that there are more constraints to the value of q as well, but the value of q will definitely be a subset to the set of rational numbers. Always remember that operations like addition, subtraction, multiplication and division between a rational and an irrational number always yields an irrational number, rational numbers involved in division and multiplication are not zero.