6. Five men collected several coconuts from a deserted island, and they decided to divide the coconuts the next day. During the night, one of them decided to separate his part so he divided the total in 5 groups and give the left-over coconut to a nearby monkey, the man went to sleep. Next, another man did the same with the remainders coconuts and give the left-over coconut to the monkey. During the night, every man did the same. Next morning the men divide the remainder coconuts and give the left-over coconut to the monkey. Question: What is the minimum number of original coconuts?
7. I have weights that weight power of 2 kilograms, like 1, 2, 4, 8, 16,.....kilograms. Which weights are necessary to weight 213 kilograms?
6. In order to determine the minimum number of original coconuts, we need to work backwards through the process described.
Let's start with the five groups that were divided on the final morning. Since they were able to divide the remaining coconuts evenly, each group must have had a multiple of 5 coconuts. Let's call the number of coconuts in each group "x."
Now, let's rewind to the previous night when every man divided their share and gave the leftover coconut to the monkey. At this point, there are 5*x + 1 coconuts, which means that the previous night's coconuts were of the form 5*x + 1.
If we go further back to the previous day, the coconuts were divided in the same way. So, we can express the number of coconuts on the previous day as 5*(5*x + 1) + 1, which simplifies to 25*x + 6.
By repeating this process for each night, we can continue working backwards until we find a value of "x" that satisfies the conditions. Ideally, we want to find the smallest value of "x" that makes the total number of coconuts a non-negative integer.
We can start with small values of "x" and keep increasing it until we reach a solution. By doing this, we can determine the minimum number of original coconuts.
7. To determine which weights are necessary to weigh 213 kilograms using weights that are powers of 2, we can use a binary representation of the desired weight.
In binary representation, every power of 2 corresponds to a bit in the binary number. Starting from the leftmost bit, a "1" indicates that the corresponding weight is needed, while a "0" indicates that it is not.
For example, to represent 213 kilograms in binary, we convert it to: 11010101.
Now, let's identify the powers of 2 for each bit position:
- The leftmost bit represents 2^7 which is 128 kilograms.
- The next bit represents 2^6 which is 64 kilograms.
- The next bit represents 2^5 which is 32 kilograms.
- The next bit represents 2^4 which is 16 kilograms.
- The next bit represents 2^3 which is 8 kilograms.
- The next bit represents 2^2 which is 4 kilograms.
- The next bit represents 2^1 which is 2 kilograms.
- The rightmost bit represents 2^0 which is 1 kilogram.
By adding up the necessary weights based on the binary representation, we can determine the combination of weights needed to weigh 213 kilograms. In this case, the necessary weights are: 128 kilograms, 64 kilograms, 32 kilograms, 16 kilograms, 4 kilograms, and 1 kilogram.