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Modular Arithmetic

Divisibility on set A

(Note: '$a|b$' read as 'a divides b')

• If A = $\mathbf{ℕ}$ ( The set of all the natural numbers)
Divisibility on $\mathbf{ℕ}$:
• If A = $\mathrm{ℤ}$( The set of all the integers)
Divisibility on $\mathrm{ℤ}$

Division Algorithm

Greatest Common Divisor (GCD) Or Highest Common Factor (HCF)

A positive integer 'd' is the GCD of a and b means 'd' is the greatest among all the common divisors of a and b i.e.
if c is any other common divisor of and b then c < d , in fact $c|d$.

In other words:
A positive integer 'd' is the GCD of a and b, if
(i)
(ii)

Note: GCD of integers a and b is denoted by (a, b).

Prime Number

A positive integer is called a prime number if its only divisors are .

Note: The smallest divisor (> 1) of an integer (> 1) is a prime number.

Coprime Numbers

If  i.e., two positive integers are co-prime iff their GCD is 1.

Some important properties of prime numbers:

Property 1):

If p is a prime number and a is any integer then (pa) = 1 or, (pa) = p.

Property 2):
Let .

Property 3):
If p is a prime number and ab are two integers then
Modular Arithmetic is a system of arithmetic for integers, …

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