Class 10 Maths- Fundamental Theorem of Arithmetic

Fundamental Theorem of Arithmetic:

The basic Idea is that any integer above 1 is either a Prime Number, or can be made by multiplying Prime Numbers together.

To understand Fundamental theorem of arithmetic, we need to first have a look at prime and composite numbers.

Prime numbers: 2, 3, 5, 7 etc.

Composite numbers:   4 or (2 x 2) , 6 or (3 x 2), 8 or ( 2 x 2 x 2) etc.

We can now define Fundamental theorem of arithmetic as:

Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique except for the order in which the prime factors occur.

Let ‘p’ be a prime number and ‘a’ be a positive integer. If p divides a², then p divides a.

Example:

Express positive integer 140 as the product of its prime factors:

140 = 2 × 2 × 5 × 7 = 2² × 5 × 7

Let's build up the ideas piece by piece:

"Any integer greater than 1" means the numbers 2, 3, 4, 5, 6, ... etc.

A Prime Number is a number that cannot be evenly divided by any other number (except 1 or itself).

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... (and more)

"...product of prime numbers" means that you multiply prime numbers together.

So, by multiplying prime numbers you can create any other whole number.

Example:

Can we make 42 by multiplying only prime numbers? Let's see:

2 × 3 × 7 = 42

Yes, 2, 3 and 7 are prime numbers, and when multiplied together they make 42.

Try some other examples for yourself. How about 30? Or 33?

It is like the Prime Numbers are the basic building blocks of all numbers.

"... unique product of prime numbers" means there is only one (unique!) set of prime numbers that will work

Example: we just showed that 42 is made by the prime numbers 2, 3 and 7:

2 × 3 × 7 = 42

No other prime numbers would work!

You could try 2 × 3 × 5, or 5 × 11, and none of them will work:

Only 2, 3 and 7 make 42