Fundamental Theorem of Arithmetic

Around 300 BC, a philosopher known as Euclid of Alexandria understood that all numbers could be split into these two distinct categories. He realized that any number can be divided down over and over until you reach a group of the smallest equal numbers. And by definition, these smallest numbers are always prime numbers. All numbers are built out of smaller primes.

Take any composite number, ex: 125, and break it down, it will always be left with prime numbers i.e. 5 × 5 × 5. Euclid knew that every number could be expressed using a group of smaller primes. No matter what number you choose, it can always be built with the addition of smaller primes. This is the root of his discovery, known as the fundamental theorem of arithmetic. Every possible number has one and only one prime factorization.

Every composite number can be written as the product of the powers of primes. For example, 4 is a composite number that can be split into 2 × 2. Similarly, 9, 18, 21, 33, and 55 could be split into 3 × 3, 3 × 3 × 2, 7 × 3, 11 × 3, and 11 × 5, respectively.

Let's consider an example 4ⁿ, where n is a natural number. Check whether there is any value for which 4ⁿ ends with the digit zero. If 4ⁿ has to end with the digit zero, then its factors must be 5 × 2. Therefore, 4ⁿ = k × 5 where k is a constant.

4ⁿ = (2 × 2)ⁿ i.e. 2 × 2 × 2 × 2 ×......×n

This does not have 5 in the prime factors. Therefore, 4ⁿ doesn't end with 0.

Let us learn how to find LCM and HCF by the prime factorisation method. For example, what are the LCF and HCF of 96 and 404 by prime factorisation?

96 = 2 × 2 × 2 ×2 × 3 Therefore 96 = 25 × 3

404 = 2 × 2 × 101 Hence 404 = 22 × 101

While finding HCF, we will take the lowest common prime factor, whereas to find LCM, the highest prime factors are considered.

HCF = 2² = 4

LCM= 25 × 3 × 101 = 9696