Rational | A real number r is rational
provided that there are integers m and n, with n0,
such that r = m/n. It can be written as:
r is rational integers m and n such that r = m/n and n0. |
There are several properties for rational numbers:
1. Every integer is a rational number.
2. The sum of any two rational numbers is rational.
3. The double of rational numbers is rational.
Any real number which is not rational is irrational. (For example, pi and the square root of 2 are irrational.)
D E F I N I T I O N S
Divides (Divisibility) | For all integers m and n, m
divides n, if and only if, there exists an integer p
such that n = mp. The notation for m
divides n is
m | n. |
Factorization Form in Prime Factors |
Given any integer m > 1, the factorization form in
prime factors of m is an expression of the form
m = p_{1}^{e1} * p_{2}^{e2} * p_{3}^{e3} ....... * p_{k}^{ek} where k is a positive integer; p_{1}, p_{2}, ........, p_{k} are prime numbers; e1, e2, ........, ek are positive integers; and p_{1} < p_{2} < ..... < p_{k}. |
Properties of Divisibility
1. Transitivity of Divisibility
For all integers x, y, z, if x divides y and y divides z, then x divides z.
2. Divisibility by a Prime Number
Any integer x > 1 is divisible by a prime number.
Let n be an integer and d be a positive integer, then there exist some integers q and r such that,
n = d * q + r and 0 < r < d
Here, q is the quotient and r is the remainder.
For instance, when dividing 25 by 4, the quotient is 6 and the remainder is 1.
Substituting these values into the above equation:
25 = 4 * 6 + 1 where x = 25, n = 4, q = 6 and r = 1.
D E F I N I T I O N S
div | Given two positive integers x and n,
x div n = the quotient obtained when x is divided by n. |
mod | Given two positive integers x and n,
x mod n = the remainder obtained when x is divided by n. |