A binary code is used to represent the days of the month. What is the shortest wordlenght necessary?

Answer 1

What's the longest length of a month?

Try 2^1, 2^2, 2^3 until you get a number larger than the length of a month. That will tell you how many bits you need.

In this case, 2^5 is 32, so you need 5 bit words to represent the days of the month.

Answer 2

Maximum value = 31: 31 = 2^5-1 so you need 5 bits
11111 = 1+2+4+8+16 = 31

Answer 3

A five-bit "nybble" could do this nicely. This will allow you to count any number from 1 to 31 exactly. For example, the 1st of the monthy would code as 00001, and the 21st as 10101. The 31st will be all ones as 11111.
The months from 1 to 12 will take a four-bit nybble, with December coding as 1100.

Answer 4

You need to be able to express 1-31 so you are under 32 choices. The number of bits needed to express 32 choices is 5.