Add $10_7 + 163_7.$ Express your answer in base 7.
To add numbers in different bases, I need to align the digits according to their place values and then add each column from right to left, just like I would in base 10. However, I also need to remember that in base 7, the digits are 0, 1, 2, 3, 4, 5, and 6, and there is no 7 or higher. So if I get a sum that is 7 or more in any column, I need to carry over a 1 to the next column and subtract 7 from the sum. For example, if I have 6 + 3 in base 7, that is 9 in base 10, but in base 7 I need to write 2 and carry over a 1. I start by adding the ones column: 3 + 0 = 3. So I write 3 in the ones column of the sum. Then I move to the sevens column: 6 + 1 = 7. But 7 is not a valid digit in base 7, so I need to carry over a 1 and write 0 in the sevens column of the sum. Finally, I move to the forty-nines column: 1 + 0 = 1. But I also have a 1 carried over from the previous column, so I add them: 1 + 1 = 2. So I write 2 in the forty-nines column of the sum. I have no more columns to add, so I am done. The final sum is 203_7. 