Question

Factorise 840. How does it justify the three conditions of fundamental theorem of arithmetic, i.e. composite or not, uniqueness and order of factors? Explain.

Answer 1

Hey mate !!

Here's the answer !!

840 = 2 × 2 × 2 × 3 × 5 × 7

So 840 = 2³ × 3 × 5 × 7

So let's come to Fundamental Theorem Of Arithmetic.

Fundamental Theorem Of Arithmetic states that:

1. All composite numbers are just a multiplication of prime factors

2. Every composite number has only one unique way of representing them in the form of product of primes.

3. This way of representation is unique for a single composite number.

So coming to the proof part.

We know that, 840 just has only one form of representing in the product of primes. That is in the form 2³ × 3 × 5 × 7. 

This order doesn't change for that particular composite number. And also this formation is possible only for this composite number and is unique for this number. Hence we can say that the formation of this product of primes is unique for a composite number and also the order of factors is same.

Hence we can justify the three statements under a single heading The Fundamental Theroem Of Arithmetic.

Hope my answer helps !!

Cheers !!