Question

The HCF and LCM of the numbers 780 and 210 are respectively
A) 30, 5460
B) 60, 2730
C) 90, 10920
D) None of these

Answer 1

Hint: We are given the two numbers, by writing down the factors by the product of which these numbers are obtained, we can find their HCF and LCM respectively. HCF is the product of common factors whereas LCM the product of all factors where common factors are counted once.

Complete step by step solution:
HCF: The full form of HCF is the highest common factor. It is the largest positive integer which divides both the numbers exactly without leaving any remainder. This can be found as the product of common factors of both the numbers.
LCM: The full form of LCM is the lowest common multiple. It is the least positive integer which is divisible by both the numbers. This can be found by finding the product of all the factors of both the numbers where the common ones are counted once only.
The factors of both780 and 210 are:
$
\Rightarrow 780 = 2 \times 5 \times 2 \times 3 \times 13 \\
\Rightarrow 210 = 2 \times 5 \times 3 \times 7 \;
 $
The common factors among the two are 2, 5, 3
The HCF is given by the product, so:
$
\Rightarrow HCF \to 2 \times 5 \times 3 \\
\Rightarrow HCF = 30 \;
 $
The LCM is given by the product of these and also the rest of the numbers left, so, it is give as:
$
\Rightarrow LCM \to 2 \times 5 \times 3 \times 2 \times 7 \times 13 \\
\Rightarrow LCM = 5460 \;
 $
Therefore, The HCF and LCM of the numbers 780 and 210 are 30 and 5460
So, the correct answer is “Option A”.

Note: When we write the factors that constitute the numbers, we always write the minimum values. For example, instead of writing 4, we will write $2 \times 2$.
To check if the answer we got as LCM and HCF is correct we can use the following formula:
Product of two numbers = Product of their HCF and LCM.
$
  780 \times 210 = 30 \times 5460 \\
  163800 = 163800 \;
 $
Thus, the answer we reached is correct.