Question

Using the fundamental theorem of arithmetic, find the LCM of $85$ and $51$.

Answer 1

Hint: For this question first we have to know about the fundamental theorem of arithmetic then find the factors of given numbers by applying prime factorization method and at last take common numbers.

Complete step-by-step answer:
According to the Fundamental Theorem of Arithmetic, every integer greater than 1 either is prime itself or is the product of a unique combination of prime numbers.
So that the prime factors of given number are
$85=17\times 5$
51=15X3
Now , we have to find the LCM. LCM of two integers or numbers is the smallest positive integer that is perfectly divisible by both numbers.
To find LCM we have to take each number from the factors, if a number is in pair then we will take one from them.
Therefore LCM of $\left( 85,51 \right)$=$17\times 3\times 5$$=255$
If we observe the factors of $85$ and $51$ are prime numbers , it means we follow the statement of Fundamental Theorem of Arithmetic .
Additional information: Prime Number- A number that is divisible only by itself and $1$.
For example: $17=17\times 1$
 If we notice, $17$is divisible by only$1$ and $17$.
Composite number- Composite numbers are which numbers that are divisible more than three numbers.
For example:$24=2\times 2\times 2\times 3$

Note: Sometimes students get confused between LCM and HCF. HCF is the highest common factor, like if we have to find HCF of $85\, and \,51$ then we take common as which number that will be present in both numbers.
Hence the HCF of $85\,and\,51$ is 17.