. 
is the numerator and 
is the
denominator. A fraction is reducible if the numerator and the denominator have a
GCD larger than 1; a fraction is irreducible if the numerator and the
denominator are coprime (their GCD is 1).A fraction is a numerical
representation that is the quotient of two numbers, . is the numerator and is the
denominator. A fraction is reducible if the numerator and the denominator have a
GCD larger than 1; a fraction is irreducible if the numerator and the
denominator are coprime (their GCD is 1).
Rational numbers are
numbers that can be expressed as the quotient of two integers: 
, where 
and
are integers. When and are
relatively prime, is
said to be the reduced form. Irrational numbers are
real numbers that cannot be written as the quotient of two integers.
If 
, 
, where 
is any number that makes
.
Problem 2‑21
If 
and 
has the same value for
any 
that satisfies 
what are
the values of 
and 
?
Solution:
If has the same value for any that satisfies 
Since 
we
have 
and 
.
Problem 2‑22
Prove that the fraction 
is irreducible for every
natural number 
Solution: This is
the first problem of the first International Mathematical Olympiad in 1959. We
know that for to be irreducible, the
GCD of the numerator 
and the denominator 
must
be 1. How do we show that? Let’s use the Euclidean algorithm.
Let’s go through the equations
in detail: When is divided by , the
quotient is 1, and the remainder is 
When is
divided by 
the quotient is 2, and
the remainder is 1. The GCD of 1 with any integer is 1. Therefore, the GCD of
the numerator and the denominator is 1,
and is irreducible.
Problem 2‑23
Solution: For both equations, we can covert the terms in the original equations to more trackable terms:
Problem 2‑24
Solution: This problem may look a little daunting, but we can follow the same logic as the last problem to simplify the original terms. All terms have the same format:
Each term in the equation is 
and
we have:
