Usually, from our knowledge of the rules of arithmetic addition, 13 + 13 = 26. So it is proper to question the validity of an expression such as 13 + 13 = 2. The expression 13 + 13 = 2 is not valid in the absence of any other definitions that are different from those used to establish the real number line. However, if some other definitions and rules are provided to change the meaning of 13 within a given mathematical situation, the expression 13 + 13 = 2 can be valid. In number theory, the definitions that allow this equality to be valid are normally found under the topic known as congruences.
The primary objective of the subject of congruences is to create a new number line, such that consecutive numbers are one unit apart and obey basic arithmetic rules, and its zero is a non-zero number of the real number line. This zero is called modulus or (mod. for short). Once a modulus is selected, a mod. number line can be established by mapping real numbers onto it. The result of this mapping is that only zero and numbers that are greater than zero and less than the modulus can be members of the mod. number line.
Suppose the number 12 is selected as the modulus, then 12 on the real number line is equivalent to zero on the mod. number line. Also all multiples of 12 on the real number line are equivalent to zero on the mod. number line. The remaining numbers on the mod. number line range from 1 through 11. For example, the number 13 = 1 on the mod. number line because 12 + 1 = 0 + 1, since 12 is the modulus. Consequently, 13 + 13 = 2 on a mod. number line that has 12 as its modulus . This equivalence is formally expressed as 13 + 13 = 2 mod 12.
In essence, the numbers on the mod. number line are the remainders of the division of real numbers by the modulus of the mod. number line. Therefore, the number on the mod. number line that is equivalent to a given real number can be determined using the following equation:
r = x - km
Where r is the number on the modul number line that is equivalent to the real number x, m the modulus, and k the number of times m divides x.
The cyclic number patterns presented by a mod. number line can be used in practice in situations where there are cyclic number patterns. For example, the days of the week form a cyclic number pattern if we select 7 as our modulus. In this situation, we assign 0 to Saturday the 7th day, and assign 1, 2, 3, 4, 5, and 6 to Sunday, Monday, Tuesday, Wednesday, Thursday and Friday respectively. Consequently, we can determine the day of the week that corresponds to x, the number of days from a reference day. If we take Saturday as the reference day, x = multiples of 7 will be Saturdays. Sundays, Mondays, Tuesdays, Wednesdays, Thursdays and Fridays will be days that leave a remainder of 1, 2, 3, 4, 5 and 6 respectively, after 7 divides x.