Question

The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number.

Answer 1

Hint: Given in the question the two numbers least common multiple and highest common factor are like a, b. We know the product of them is nothing but the product of both numbers. So, in the equation of 4 variables we know 3 variables. So, we can find the value of the $4^{th}$ variable easily by just algebraic solving techniques.

Complete step-by-step solution -
By basic algebraic knowledge, we know that this is true:
L.C.M be least common multiple of a, b
H.C.F Be highest common factor of a, b
a, b any 2 numbers.
LCM×HCF= a×b
Before using we will just verify this condition by normal assumptions.
Assume, a= 26, b= 91
By doing prime factorization, we get the following as:
$\begin{align}
  & 26=2\times 13 \\
 & 91=7\times 13 \\
\end{align}$
First, we take the highest common factor of 26, 91. So, the common number which is highest will be 13.
So, HCF is 13.
Now we find the least common multiple of 91, 26.
We take the numbers which are not repeated and repeated will be grouped.
LCM of 26, 91 = $2\times 7\times 13=2\times 91$
Verify, HCF×LCM= a×b
$\begin{align}
  & 2\times 91\times 13=26\times 91 \\
 & 91\times 26=26\times 91 \\
\end{align}$
Hence verified
Here in the given question we know three variables of above condition. The highest common factor of the given two numbers is: 6. The least common factor of the given two numbers is 180.
The first number which is known to us by question is 30. The second number which is unknown to us, let it be k. By applying the condition, we know HCF×LCM=a×b.
By substituting all the values, we get:
$6\times 180=30\times k$
By dividing with 30 on both sides of the equation, we get.
$k=\dfrac{6\times 180}{30}=6\times 6$
By simplifying we get the value of k to be 36.
Therefore 36 is the value of the second number that satisfies all conditions.

Note: Be careful while applying formulas as LCM, HCF should on LHS and both numbers must be on RHS. We use the property that multiplication of HCF and LCM of two numbers is equal to the product of the numbers.