# Real World Applications of Modular Theory

### A brief overview of Modular Theory

In some situations, we are only interested in the remainder of an integer when it is divided by a positive integer. For instance, when we are asked what time it will be after 40 hours (on a 12-hour clock), we are only interested in the remainder, i.e 40 mod 12.

“If a and b are integers and m is a positive integer, then a is congruent to b modulo m if a = b + km, where k is an integer constant. The notation used is a ≡ b (mod m) to indicate that a is congruent to b modulo m.”

For example: 19 is congruent to 7 modulo 6 [19 ≡ 7 (mod 6)], but 4 is not congruent to 36 modulo 5 [4 ≢ 36 (mod 5)]. The above statement can also be written as an “if a mod m = b mod m, then a is congruent to b modulo m”. I hope this brief overview of modular arithmetic would help you to understand its applications better.

### Applications of Modular Theory :

• Parity Check Bits — Digital information is represented by bit string, split into blocks of a specified size. Before each block is stored or transmitted, an extra bit, called a parity check bit, is appended to each block. The parity check bit x[n+1] for the bit string x[1]x[2]…x[n] is defined
by “ x[n+1] = x[1] + x[2] + ….+x[n] mod 2 ”.

It follows that x[n+1] is O if there are an even number of 1 bits in the block of n bits and it is 1 if there are an odd number of 1 bits in the block of n bits. When we examine a string that includes a parity check bit, we know that there is an error in it if the parity check bit is wrong. However, when the parity check bit is correct, there still may be an error. This is used as an additional check to ensure that the packets of data received are correct and complete by the machine.

• ISBNs — All books are identified by an International Standard Book Number, a 10-digit code. A new system of 13-digit ISBN code has recently been introduced to identify a large number of books. But here our main focus would on ISBN-10. An ISBN-10 consists of blocks identifying the language, the publisher, the number assigned to the book by its publishing company, and finally the last digit is either a digit or the letter X (used to represent 10 in Roman system). This check digit is selected so
that “ ∑ i*x[i] ≡ 0 (mod 11), where 1 ≦ i ≦ 10” .

The above expressions can be simplified as “ x[10] = 1*x[1] + 2*x[2] + 3*x[3] + 4*x[4] + 5*x[5] + 6*x[6] + 7*x[7] + 8*x[8] + 9*x[9] mod(11).” The last digit also called the check digit is used to verify if there is any printing mistake in the ISBN. This is actually very interesting and you can verify yourself through the above simple calculation. In the new system ISBN-13, instead of 10, 13 digits are used and the check digit is always a digit, i.e 0–9.

Modular Theory has many more applications to discrete mathematics, computer science, and many other disciplines; like generation of pseudo-random numbers, Hashing functions, Cryptography and even in Music!