Question

If HCF of \[\left( {420,x} \right) = 60\] and LCM of \[\left( {420,x} \right) = 1260\], then \[x\] is
A.180
B.300
C.210
D.250

Answer 1

Hint: Here, we will find the number. We will use the concept of relation between H.C.F. and L.C.M. to find one of the given two numbers by using LCM and HCF of two given numbers and By equating and solving, we will find one of the two numbers. Thus the required number.

Formula Used:
The product of the least common multiple and the highest common factor of the natural numbers is always equal to the product of the natural numbers.
i.e., L.C.M. of the given numbers \[ \times \] H.C.F. of the given numbers \[ = \] Product of the given numbers.

Complete step-by-step answer:
We are given that the highest common factor (H.C.F) of the numbers \[420,x\] as \[60\].
We are given that the least common multiple (L.C.M) of the numbers \[420,x\] as \[1260\].
We are given a number as \[420\] and another number as \[x\].
Now, we will find another number by using the relation between H.C.F. and L.C.M.
We know that the product of the least common multiple and the highest common factor of the natural numbers is always equal to the product of the natural numbers.
\[ \Rightarrow \] L.C.M. of the given numbers \[ \times \] H.C.F. of the given numbers \[ = \] Product of the given numbers.
By substituting H.C.F. of the number as 60, L.C.M of the number as 1260 and one of the number as 420, we get
\[ \Rightarrow 1260 \times 60 = 420 \times x\]
By rewriting the equation, we get
\[ \Rightarrow x = \dfrac{{1260 \times 60}}{{420}}\]
By dividing the numbers, we get
\[ \Rightarrow x = \dfrac{{1260}}{7}\]
By dividing the numbers, we get
\[ \Rightarrow x = 180\]
Therefore, the number \[x\] is \[180\]
Option (A) is the correct answer.

Note: We know that the Highest Common Factor(H.C.F) of two numbers is defined as the greatest number which divides exactly both the numbers. The Least Common Multiple(L.C.M) of two numbers is defined as the smallest number which is divisible by both the numbers. We know that the H.C.F. of two co-prime numbers is 1. Thus, the least common multiple of two co-prime numbers is equal to the product of the given two co-prime numbers. So, the given numbers are not coprime numbers.