Introduction to Modular Arithmetic

Running head: SIT192
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Introduction to modular arithmetic:
The modular arithmetic is considered to be a branch of mathematics where arithmetic
operations are performed with integer numbers. It’s a cyclical process where numbers are
repeated after reaching a certain value and the repeated numbers in a sequence is called
modulus (plural is moduli) (Rosenthal, Rosenthal and Rosenthal 2018). The modular
arithmetic is developed and modernized by Carl Friedrich Gauss in 1801 in his book
“Disquisitiones Arithmeticae”.
The most familiar use of the modular arithmetic is the 12 hour clock where hours go from 1
to 12 and after that same number is repeated. Hence, 13th hour is 1 o’clock, 14th is 2 o’clock
and so on. Hence, in clock time 1,13,25... are same and so as 2,14,26,38... and so on. The
remainder when 1,13,25 are divided by 12 is 1. Mathematically this is represented as 25 ≡ 1
mod 12 or in language it is “25 is congruent to 1 mod 12” (Kim and Tibouchi 2016). There
are several properties of modular congruence that are used in large calculations are listed
below.
Properties of modular congruence:
1. if a, b and n are integers, then a will be congruent to b modulo n or mathematically a ≡ b
(mod n) iff (i.e. if and only if) (a-b) is divisible by n or mathematically n | (a-b)
2. Reflexive property of modular congruence: If n and a are integers then a is congruent to a
modulo n or a ≡ a (mod n).
3. Symmetric property of modular congruence: if n, a and b are integers and if a is congruent
to b modulo n then b is congruent to a modulo n (Kraft and Washington 2018).
Mathematically, if a ≡ b (mod n) then b ≡ a (mod n)

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4. Transitive property of modular congruence: if a, b, c and n are all integers and if a is
congruent to b modulo n and b is congruent to c mod n, then a is congruent to c modulo n.
Mathematical expression: if a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n).
5. Additive property of congruence: If a,b,c, d and n are all integers and if a is congruent to c
mod n and b is congruent to d mod n then a+b ≡ c + d mod n.
This can further be extended to relation (a+b) mod n = c+d mod n.
6. Multiplicative property of congruence: If a,b,c,d and n are all integers and if a is congruent
to c mod n and b is congruent to d mod n then ab is congruent to cd modulo n (or ab ≡ cd
mod n).
The above relation can further be extended to relation ab (mod n) = cd (mod n).
Proof of all the above properties are simple but beyond the length of this report and hence the
proof of only additive and multiplicative congruence as given in question 1 are proved with
detail explanation in the later section.
Question 1:
It is required to be proved that
If a = c mod m and b = d mod m
Then,
i) (a+b) mod m = (c + d) mod m
ii) ab mod m = cd mod m
Here, all a, b, c, d are m are all considered as positive for convenience.
Prove of i)