### Instruction: Find the product of $1011_2 \cdot 101_2$.  Express your answer in base 2.### Response: I know that $1011_2$ is equivalent to 8 + 2 + 1 = 11 in base 10, and $101_2$ is equivalent to 4 + 1 = 5 in base 10.
So the product of these two numbers in base 10 is 11 * 5 = 55.
To convert 55 to base 2, I can use repeated division by 2 and write down the remainders.
55 / 2 = 27 with remainder 1, so the rightmost digit of the base 2 representation is 1.
27 / 2 = 13 with remainder 1, so the next digit is 1.
13 / 2 = 6 with remainder 1, so the next digit is 1.
6 / 2 = 3 with remainder 0, so the next digit is 0.
3 / 2 = 1 with remainder 1, so the next digit is 1.
1 / 2 = 0 with remainder 1, so the final digit is 1.
Therefore, 55 in base 2 is $110111_2$.