Question

Given that HCF (26, 91) =13, then the LCM of (26, 91) is:
(a) 2366
(b) 182
(c) 91
(d) 364

Answer 1

Hint: First look at definitions of highest common factor, least common multiple. Then by using relation between them find the relation of product. First verify the relation. Now use the given condition to find the result. You have 4 variables, 2 numbers, their highest common factor, their least common multiple. You already know 3 variables by finding the value of remaining variables.

Complete step-by-step answer:
By basic algebraic knowledge, we know that this is true:
LCM be least common multiple of a, b
HCF be highest common factor of a, b
a, b be any two numbers.
$LCM\times HCF=a\times b$
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:
$\begin{array}{rl}& 26=2\times 13\\ & 091=7\times 13\end{array}$
First, we take the highest common factor of 26, 91. So, the common number which is highest will be 13.
So, HCF=13
Now, we find LCM 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, $HCF\times LCM=a\times b$
$\begin{array}{rl}& 2\times 91\times 13=26\times 91\\ & 91\times 26=26\times 91\end{array}$
Hence verified
Here in given question we know three variables of above condition: the highest common factor of the given two numbers is: 13
The least common factor of the given two numbers is an unknown unit called LCM. The first number which is known to us by question is 26. The second number which is known to us by question is 91.
Given condition in the question can be written in the form:
$HCF(26,91)=13$
By this one equation, we can say the value 3 variables. HCF, a, b, as:
HCF=13, a=26, b=91
By substituting these values into the equation, we get it as:
$13\times LCM=26\times 91$
By simplifying the above equation, we write the equation as:
$13\times LCM=2366$
By dividing with 13 on both sides of equation, we get:
$LCM={\displaystyle \frac{2366}{13}}=182$
Therefore option (b) is the correct answer.

Note: Be careful as the pair should be LCM, HCF on one side and both numbers on the other side. verify the equation properly. While simplifying be careful that you might not miss any value this may lead to wrong results.