Question

Find the HCF and LCM of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 510 and 92

Answer 1

Hint: HCF is the highest common factor of the two or more numbers while LCM is the lowest common multiple of the given two or more numbers.
HCF is less than the LCM of a given pair of numbers.

Complete step by step solution:
To find the factors of $510$ and $92$, $510$ and $92$ factorized using the prime factorization method. Prime factorization method takes prime factors (numbers which are only divisible by one and itself) to determine factors of a given number.
Factors of $510$ are: $510=2\times 3\times 5\times 17\times 1$
Factors of $92$ are: $92=2\times 2\times 23$
The highest common factor of numbers: $510$ and $92$ are determined by taking the common factors of $510$ and $92$. Common factors of $510$ and $92$ are: $2$ and $1$. This indicates HCF of $510$ and $92$= $2\times 1$ = $2$
The lowest common multiple of two numbers $510$ and $92$ is determined by taking the common factor of two given numbers as well as the remaining non-common factors.
LCM of $510$ and $92$ is: $LCM=2\times 3\times 5\times 17\times 2\times 23=23,460$.
To verify that HCF × LCM is equal to product of numbers, $510$ and $92$, the LHS and RHS of this identity needs to be calculated which is as follows:
HCF × LCM = $2\times 23,460=46,920$
Product of numbers $510$ and $92$ = $510\times 92=46,920$
As both Left-hand side and Right-hand side of identity are equal to each other, the identity proved.

Note:
If one of the two numbers is given and their LCM and HCF are known, another number can be determined using the above identity.