When the base-10 integers 200 and 1200 are expressed in base 2, how many more digits does 1200 have than 200 (after being converted)?
I know that to convert a number from base 10 to base 2, I need to repeatedly divide it by 2 and write down the remainders, until I get a quotient of 0. For example, to convert 200 to base 2, I start by dividing it by 2 and getting a quotient of 100 and a remainder of 0. Then I divide 100 by 2 and get a quotient of 50 and a remainder of 0. I continue this process until I get a quotient of 0 and a remainder of 1. The remainders form the digits of the base 2 representation, starting from the rightmost one. So, 200 in base 2 is 11001000, which has 8 digits. Similarly, to convert 1200 to base 2, I start by dividing it by 2 and getting a quotient of 600 and a remainder of 0. Then I divide 600 by 2 and get a quotient of 300 and a remainder of 0. I continue this process until I get a quotient of 0 and a remainder of 1. So, 1200 in base 2 is 10010110000, which has 11 digits. Therefore, 1200 in base 2 has 11 - 8 = 3 more digits than 200 in base 2. 