correct answer is: c) 13711

Explanation

First, we find the LCM of the given number 152 and 285.
The LCM of 152 and 285 is =19\times8\times15.
\rightarrow 760.
Now, if we multiply 760 by a certain number, the product of which falls between 13000 and 14000, we get :
\rightarrow\left(760\times18\right)=13680.
So, 13680 is the number between 13000 and 14000 that is divisible by both 152 and 285.
And, if we add 31 to 13680 then we get =\left(13680+31\right)
\rightarrow 13711.
\therefore 13711 is the number between 13000 and 14000 that, when divided by 152 and 285, leaves a remainder of 31 in each case.
Ans: The required number between 13000 and 14000 is 13711.

Another method to articulate this particular math query is available:

1. Identify the number between 13000 and 14000 that yields a remainder of 31 when divided by 152 and 285 respectively.
2. What is the integer within the interval of 13000 to 14000 that leaves a remainder of 31 when divided by 152 and 285?