Fundamental Theorem of Arithmetic

In this section of fundamental theorem of arithmetic we will learn about its various applications . For example, we have used prime factorization method to find GCF (HCF) and LCM of positive integers. In this method, we use the fundamental arithmetic theorem in expressing the given integers as the product of primes.
Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers (In fact, infinitely many). Let us list a few :
1771 = 7 × 11 × 23
8232 = 2 x 2 x 2 x 3 x 7 x 7 x 7
∴ 8232 = 2³ x 3 x 7³ and so on.

Theorem (Fundamental Arithmetic Theorem) : Every composite number can be expressed ( factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic says that every composite number can be factorized as a product of primes.
It says that given any composite number it can be factorized as a product of prime numbers in a ‘unique’ way, except for the order in which the primes occur. That is, given any composite number there is one and only one way to write it as a product of primes, as long as we are not particular about the order in which the primes occur.
So, for example, 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:
The prime factorization of a natural number is unique, except for the order of its factors.

Examples :
1) Find the LCM and HCF of 6 and 20 by the prime factorization method.
Solution :
We have : 6 = 2 × 3
20 = 2 × 2 × 5
20 = 2² × 5.
Common factors of 6 and 20 are 2¹ and 2²
So for HCF take the common number with lowest exponent.
∴ HCF = 2¹ = 2
In LCM take the common factor with highest exponent and the remaining factors.
∴ LCM of 6, 20 = 2² × 3 × 5
∴ LCM = 4 x 3 x 5 = 60
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2) Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.
Solution :
6 = 2 × 3,
72 = 2³ × 3²
120 = 2³ × 3 × 5
Common factors are 2, 2³, 3, 3²
So for HCF take the common number with lowest exponent.
∴ HCF = 2 x 3 = 6
In LCM take the common factor with highest exponent and the remaining factors.
∴ LCM of 6,72 and 120 = 2³ x 3² x 5
∴ LCM = 8 x 9 x 5 = 360.

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Euclid's Geometry

• Euclid Geometry
• Euclids division lemma
• Euclids division Algorithm
• Fundamental Theorem of Arithmetic
• Finding HCF LCM of positive integers
• Proving Irrationality of Numbers
• Decimal expansion of Rational numbers

From fundamental theorem of arithmetic to Real numbers

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