The fundamental theorem of arithmetic
Natural numbers are divided into prime numbers and composite numbers.
A prime number is a Natural number greater than 1 whose only factors are 1 and itself.
A composite number is a Natural number that has factors other than 1 and itself.
Consider a composite number
$210$
for factorisation.
So we could write
$210$
as the product of prime numbers
$2,3,5$
and
$7$
.
We can express any number as a product of prime numbers.
Let us verify this with some more examples
Let's factorise
$7875$
using factor tree.
So,
$7875$
can be written as the product of primes
$3,5$
and
$7$
Let us now factorize
$22230$
using factor tree.
So,
$22230$
can be written
as the product of primes
$2,3,5,13$
and
$19$
.
So, we can say that every composite number can be written as a product of primes.
This statement is true and is called the fundamental theorem of arithmetic.
The fundamental theorem of arithmetic
It states that every composite number can be expressed as a product of prime numbers.
This factorisation is unique, irrespective of the order of factorisation.
Let's solve an example using the fundamental theorem of arithmetic
Check if there is a natural number
$n$
for which the number
$4_{n}$
ends in zero.
If the number
$4_{n}$
ends in zero, then it would be divisible by
$5$
So, the prime factorization of
$4_{n}$
should contain the prime
$5$
This is not possible because
$4_{n}$
=
$(2)²ⁿ$
, and the only prime in the factorisation of
$4_{n}$
is
$2$
From the fundamental theorem of arithmetic, we know that the prime factorization of a number is unique.
So, the fundamental theorem of arithmetic guarantees that there are no other primes in the factorisation of
$4_{n}$
So, there is
no natural number
$n$
for which
$4_{n}$
ends in zero.
Let's take another example
Asha takes
$20$
minutes to ride around the park.
Whereas Cole takes
$15$
minutes to complete one round around the same park.
After how much time will they meet again at starting point if both start at the same time.
We can find LCM using the fundamental theorem of arithmetic.
So, they meet again at the starting point after
$60$
minutes, i.e.
$1$
hour.
Time to recap
The fundamental theorem of arithmetic states that every composite number can be written as a product of primes.
This factorisation is unique, irrespective of the order of factorisation.
Keep learning.