In a given month, some days are more similar than others. For example, the 3rd of the month is more like the 24th than it is like the 15th. What does this possibly mean? We will explore two ways in which this is true.
We will say that two numbers between 1 and 31 are related, written \(a \sim b\) if their difference is a multiple of 7. So for example, \(3 \sim 24\text{,}\) since \(24-3 = 3\cdot 7\text{,}\) but \(3 \not\sim 15\) since \(15-3 = 12\) which is not a multiple of 7.
When you divide a multiple of 7 by 7, you get a whole number. If you divide another number by 7, you can either write the result as a decimal or as a quotient and a remainder. For example, \(19 \div 7\) is \(2\) with a remainder of 5, since we can write \(19 = 2\cdot 7 + 5\text{.}\) The remainder is also called the modulus. When programming in python (and many other languages), the modulus operator is written as %. For example, 19 % 7 is 5. Try this out for a few numbers.
Since the modulus is a function, each number has exactly one modulus when divided by 7. This means that the moduli partition the numbers from 1 to 31: every number belongs to exactly one of the sets of numbers with a particular modulus. We have already found the set for modulus 5. Find the other sets.
We can use the moduli to define a relation on the numbers from 1 to 31. We will say that \(a \sim b\) if a % 7 = b % 7. In other words, two numbers are related if they belong to the same set of the partition we found above.
