Question

A position integer of the form 24q+9 can also be written as (m, q are integers):
\[\begin{align}
  & \text{A}.\text{ 8m} \\
 & \text{B}.\text{ 8m}+\text{2} \\
 & \text{C}.\text{ 8m}+\text{1} \\
 & \text{D}.\text{ 8m}+\text{7} \\
\end{align}\]

Answer 1

Hint: In the above question, first of all we will write 9 from the given positive integers as the sum as two positive integers such that at least one will be multiple of 8 and then, we will take 8 as common from the given form of the positive integer.

Complete step-by-step answer:
We have been asked to write a positive integer of the form 24q+9 in another form. Also, it is given that q is an integer.
Now, in the positive integer of the form 24q+9, we will write the term 9 as the sum of two positive integers such that, at least one will be the multiple of 8.
\[\begin{align}
  & \Rightarrow \text{9 can be written as }\left( 8+1 \right) \\
 & \Rightarrow 24q+9=24q+8+1 \\
\end{align}\]
On taking 8 as a common from the possible terms, we get,
\[\Rightarrow 8\left( 3q+1 \right)+1\]
We have been given that q is an integer.
\[\Rightarrow \left( 3q+1 \right)\text{ is also an integer}\text{.}\]
Let us suppose (3q+1) be m, where m is also an integer.
\[\Rightarrow 24q+9=8m+1\]
Hence, the given positive numbers can also be written in the form of 8m+1.
So, the correct answer is “Option C”.

Note: We can also find the correct option by checking the option by substituting some numerical value. It will be very time consuming. If we take q=1, we get 24+9=33. If we take m=1 and check each option, we get 8, 10, 9 and 15. This is not possible, for we take m=2 and check. We will get 16, 18, 17 and 23. Now, let us check for m=3, so we get 24, 26, 25 and 31. For m=4, we get 32, 34, 33 and 39. Now we can see that the option 8m+1 gives 33 as the answer. So, this method is very hectic and I do not use this.