What is the sum of all the positive numbers smaller than 100,000 where all digits can only be 1 or 0.
To find the sum of all the positive numbers smaller than 100,000 where all digits can only be 1 or 0, we can use a simple mathematical approach.
Let's break down the problem into smaller steps:
1. Determine the maximum number of digits allowed in each number.
2. Generate all possible numbers within the given constraints.
3. Add up all the generated numbers to find the sum.
Step 1: Determine the maximum number of digits allowed.
Since we want to find numbers smaller than 100,000, with digits restricted to 0 or 1, let's find the maximum number of digits allowed. To do this, we need to find the highest power of 2 that is smaller than 100,000.
Starting from 2^0, we find:
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
The highest power of 2 that is smaller than 100,000 is 2^16, which equals 65,536. Hence, the maximum number of digits allowed is 16.
Step 2: Generate all possible numbers within the given constraints.
To generate all possible numbers, we need to consider each digit's position, starting from the rightmost digit (the units place) up to the leftmost digit (the highest place value). Each position can have either 0 or 1 as a digit.
For example, with 16 digits allowed, the first number we can generate is 0000000000000001, followed by 0000000000000010, then 0000000000000011, and so on, up to the largest number 1111111111111111.
Step 3: Add up all the generated numbers to find the sum.
Once we have generated all possible numbers, we can add them up to find the sum. Since each digit is restricted to 0 or 1, the leftmost digit always represents the power of 2 corresponding to the number's position.
To calculate the sum, we can use the following formula:
Sum = (2^0 * 1) + (2^1 * 1) + (2^2 * 1) + ... + (2^15 * 1) + (2^16 * 1)
Using the formula, we can compute the sum.
Note: Since there are a large number of possible numbers, calculating the exact sum manually might be time-consuming. Using a computer program or a calculator is recommended for this kind of calculation.