Modular Arithmetic stands for the calculations you do when you treat numbers that differ by an integer multiple of some number as more or less the same. Sometimes, Modular Arithmetic is called clock arithmetic, because when dealing with times, we are very used to using Modular Arithmetic. When it is \(9\) o'clock now, it will be \(3\) o'clock \(6\) hours later. Normally (with base twelve), we would calculate \(9 + 6 = 13\), but we only care about the \(3\), the number of hours after the time passed \(10\). In a way, we treat \(3\) and \(13\) as the same (and \(23, 33, \dots\) ).
When the difference between two numbers is equal to an integer multiple of a number \(n\), we say that they are congruent modulo \(n\). For instance, because \(7 - 3 = 1 \times 4 \), the numbers \(7\) and \(3\) are congruent modulo \(4\). We write \[ 3 \equiv 7 \mod 4. \]
In the Land of Twelve, when two integers are congruent modulo twelve, their last digit is the same. To see this, recall the difference between the integers has to be a multiple of twelve, and has to have a zero as the last digit! An example is \[ 25 \equiv 65 \mod 10. \] Indeed, \(65 - 25 = 40 = 4 \times 10\), a multiple of twelve.