Question

Given that L.C.M \[\left( {26,169} \right) = 338\], find H.C.F \[\left( {26,169} \right)\].

Answer 1

Hint: Here, we will use the relation between H.C.F and L.C.M. We will substitute the given values in the relation and simplify it further to get the required HCF of the given numbers. H.C.F (Highest Common Factor) is the greatest number that divides each of the given integers completely. L.C.M (Least Common Multiple) is the smallest number which is a multiple of the given numbers.

Complete step-by-step answer:
The two numbers are 26 and 169 and their L.C.M is given as L.C.M \[\left( {26,169} \right) = 338\].
We will find the H.C.F of numbers using the relation it has with LCM of the same numbers.
Now we know that the formula for the relation between L.C.M (Least Common Multiple) and H.C.F (Highest Common Factor) of two numbers is given as
L.C.M \[ \times \] H.C.F \[ = \] Product of two numbers
Substituting the respective values in the above relation, we get
\[ \Rightarrow 338 \times \] H.C.F \[ = 26 \times 169\]
\[ \Rightarrow \] H.C.F \[ = \dfrac{{26 \times 169}}{{338}}\]
Multiplying the terms, we get
\[ \Rightarrow \] H.C.F \[ = \dfrac{{4394}}{{338}}\]
Dividing 4394 by 338, we get
\[ \Rightarrow \] H.C.F \[ = 13\]
So H.C.F \[\left( {26,169} \right) = 13\]

Note: H.C.F of a co-prime number is always equal to 1 as there will be no common number as there divisor whereas L.C.M of co-prime numbers is always equal to the product of the two numbers. H.C.F contains all the common factors of two numbers and L.C.M contains all the factors irrespective of common or uncommon. So, the two of them combined have all the factors common in double and non-common in single. Thus, the product of two numbers is equal to the product of their least common multiple and highest common divisor.