### Instruction: What is the base-10 integer 515 when expressed in base 6?### Response: To convert from base 10 to base 6, I need to find the largest power of 6 that is less than or equal to 515, and then use that as the highest place value.
6^3 = 216, which is less than 515, but 6^4 = 1296, which is too big.
So the highest place value is 6^3, and I need to find how many times 6^3 goes into 515.
I can use long division or repeated subtraction to find that 515 / 216 = 2 with a remainder of 83.
So the coefficient of 6^3 is 2.
Next, I need to repeat the process with the remainder of 83 and the next lower power of 6, which is 6^2 = 36.
I find that 83 / 36 = 2 with a remainder of 11.
So the coefficient of 6^2 is also 2.
Then, I do the same with 11 and 6^1 = 6.
I get 11 / 6 = 1 with a remainder of 5.
So the coefficient of 6^1 is 1.
Finally, I have 5 as the remainder and 6^0 = 1 as the lowest power of 6.
The coefficient of 6^0 is just the remainder itself, which is 5.
So, I have found the coefficients of each power of 6 from highest to lowest: 2, 2, 1, 5.
To write the base-6 number, I just concatenate these coefficients in the same order: 2215.
Therefore, 515 in base 10 is 2215 in base 6.