Final answer:
The question involves finding numeral bases a and b that satisfy the equation 31a = 25b. This requires understanding positional numeral systems and solving for integer values a and b that make the equation true.
Step-by-step explanation:
The question Determine the base(s) in which 31a = 25b is asking to solve for the numeral system (or bases) in which the equation holds true. It requires an understanding of how numbers are represented in different bases and how to convert between numeral systems.
To determine the appropriate bases, we compare the place values of the digits in each positional numeral system. For base a, the digit '3' would have a place value of 3 times a to the first power, and the digit '1' would have a place value of 1 times a to the zeroth power. Similarly, for base b, the digit '2' would have a place value of 2 times b to the first power, and the digit '5' would have a place value of 5 times b to the zeroth power.
Formally, we can equate the base a and base b expressions:
3a1 + 1a0 = 2b1 + 5b0
This equation simplifies to:
3a + 1 = 2b + 5
Then, we must explore various values of a and b that satisfy this equation until we find integer solutions that represent legitimate numeral bases (greater than 1). We may need to use trial and error, algebraic manipulation, or number theory techniques to solve for a and b.