The integers 195 and 61 are expressed in base 4 and added.  What is the resulting sum, expressed in base 4?
First, I need to convert 195 and 61 to base 4. I can use repeated division by 4 to find the digits of 195 in base 4. 195 divided by 4 is 48 with a remainder of 3, so the rightmost digit is 3. 48 divided by 4 is 12 with a remainder of 0, so the next digit is 0. 12 divided by 4 is 3 with a remainder of 0, so the next digit is 0. 3 divided by 4 is 0 with a remainder of 3, so the leftmost digit is 3. Therefore, 195 in base 4 is 3003. 61 divided by 4 is 15 with a remainder of 1, so the rightmost digit is 1. 15 divided by 4 is 3 with a remainder of 3, so the next digit is 3. 3 divided by 4 is 0 with a remainder of 3, so the next digit is 3. Therefore, 61 in base 4 is 0331. Now, I can align the digits of 3003 and 0331 according to their place values and add them from right to left. 3 plus 1 is 4, but 4 is not a valid digit in base 4, so I need to carry over 1 to the next place and write 0 in the rightmost place. 0 plus 3 is 3, but I have to add the carried over 1, so I get 4 again, and I need to carry over 1 to the next place and write 0 in the next place. 0 plus 3 is 3, but I have to add the carried over 1, so I get 4 again, and I need to carry over 1 to the next place and write 0 in the next place. 3 plus 0 is 3, but I have to add the carried over 1, so I get 4 again, and I need to carry over 1 to the next place and write 0 in the leftmost place. I have no more digits to add, but I have a carried over 1, so I need to write 1 in the next place to the left. Therefore, the sum of 3003 and 0331 in base 4 is 10000. 