[Maths Class Notes] on Fundamental Theorem of Arithmetic Pdf for Exam

Prime Numbers and Composite Numbers

A prime number is a natural number that is greater than 1 and it is not a product of two smaller natural numbers. A composite number is a number greater than 1 that is not a prime number, which means that a particular number is a natural number and can be expressed as a product of two or more than two numbers.

Fundamental Theorem of Arithmetic 

In the year 1801, Carl Friedrich Gauss proved the fundamental theorem of Arithmetic (FTA). This theory states that every composite number can be expressed as a product of two or more prime numbers; this method of factorization is unique except for the order in which these prime factors occur. So this theory is sometimes also called a unique factorization theorem or unique prime factorization theorem.

The proof of Fundamental Theorem of Arithmetic is given below:

Let us consider a composite number 140.  Factorize this number. If we keep on doing the factorization we will arrive at a stage when all the factors are prime numbers.

So, we have 140 = 2 x 2x 5 x 7.

140 can also be expressed as, 140 = 5 x 2 x 7 x 2

Hence, we observe that in both the cases, the factorization of 140, the prime numbers which appear are the same, but the order in which they appear is different. So we see that the prime factorization of 140 is unique except the order in which the prime numbers occur.

Solved Examples

1. Express Each of the Following Positive Integers as the Product of its Prime Factors by Prime Factorization Method.

a.156  b. 234

a.  156

Solution: 156 = 2 x 78

= 2 x 2 x 39

= 2 x 2 x 3 x 13

    156 = 2 x 2 x 3 x 13

b. 234

Solution: 234 = 2 x 117

=2 x 3 x 39 

= 2 x 3 x 3 x 13

234 = 2 x 3 x 3 x 13 

2. Find the HCF and LCM of 26 and 91 and Prove that LCM × HCF = Product of Two Numbers.

Solution:

By prime factorization

26 = 2 x 13

91 = 7 x 13

HCF (26, 91) = 13

LCM (26, 91) = 13 x 2 x 7 

            = 26 × 7 = 182

LCM × HCF = 13 × 182   = 2366

Product of two numbers = 26 × 91 = 2366.

Hence, L.C.M. × H.C.F. = Product of two numbers.

Quiz Time

1. Find the HCF X LCM for the numbers 105 and 120

2. The HCF of two numbers is 18 and their LCM is 720. If one of the numbers is 90, find the other

Conclusion

We have explained a few examples above based on the concept for your better understanding. These notes on Fundamental Theorem of Arithmetic are created by subject-specific experts and academics with excellent knowledge on the topic. These notes are highly accurate, provide lucid explanations and illustrations wherever necessary, and thus provide an excellent learning material resource for the students to prepare for their exams.