Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$?
Any number in $S$ has a leftmost (eights) digit that is equal to $1$. The remaining three digits can either be $0$ or $1$, so there are a total of $2^3 = 8$ elements in $S$. Notice that element $x$ in $S$ can be paired with another element $10111_2-x$, which is the base $2$ number whose right-most three digits are the opposite of those of $x$. Thus, the sum of the elements in $S$ is equal to $4 \times 10111_2 = 100_2 \times 10111_2 = \boxed{1011100}_2$.
1011100
Level 5
Number Theory
Problem: Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the sum of all four-digit positive integers in base 2, we first identify the range of these integers. The smallest four-digit binary number is 1000 (which is 8 in decimal), and the largest is 1111 (which is 15 in decimal). Therefore, the set S consists of the integers 8, 9, 10, 11, 12, 13, 14, and 15.

Next, we calculate the sum of these integers. The sum can be computed directly:

8 + 9 + 10 + 11 + 12 + 13 + 14 + 15.

This is an arithmetic series where the first term a = 8, the last term l = 15, and the number of terms n = 15 - 8 + 1 = 8. The sum of an arithmetic series can be calculated using the formula:

Sum = n/2 * (a + l).

Substituting the values:

Sum = 8/2 * (8 + 15) = 4 * 23 = 92.

Now, we need to express this sum in base 2. The decimal number 92 can be converted to binary. We divide 92 by 2 repeatedly and record the remainders:

92 ÷ 2 = 46 remainder 0  
46 ÷ 2 = 23 remainder 0  
23 ÷ 2 = 11 remainder 1  
11 ÷ 2 = 5 remainder 1  
5 ÷ 2 = 2 remainder 1  
2 ÷ 2 = 1 remainder 0  
1 ÷ 2 = 0 remainder 1  

Reading the remainders from bottom to top gives us 1011100 in binary.

Thus, the final answer is 1011100.<|END_OF_LATENT|> The final answer is $1011100$.