Question

The product of two numbers is 750 and the H.C.F. is 5. Find the L.C.M.

Answer 1

Hint: Here, we need to find the L.C.M. of the two numbers. We will use the property of the product of H.C.F. and L.C.M. of two numbers to form an equation. Then, we will simplify the equation to find the L.C.M. of the two numbers.

Complete step-by-step answer:
Let the two numbers be \[a\] and \[b\].
It is given that the product of the two numbers is 750.
Therefore, we get
\[a \times b = 750\]
The H.C.F. of the two numbers \[a\] and \[b\] is given by \[{\rm{H}}{\rm{.C}}{\rm{.F}}{\rm{.}}\left( {a,b} \right)\].
It is given that the H.C.F. of the two numbers is 5.
Therefore, we get
\[{\rm{H}}{\rm{.C}}{\rm{.F}}{\rm{.}}\left( {a,b} \right) = 5\]
The L.C.M. of the two numbers \[a\] and \[b\] is given by \[{\rm{L}}{\rm{.C}}{\rm{.M}}{\rm{.}}\left( {a,b} \right)\].
The product of the H.C.F. and L.C.M. of two numbers \[a\] and \[b\] is equal to the product of the two numbers \[a\] and \[b\].
Thus, we get the equation
\[{\rm{H}}{\rm{.C}}{\rm{.F}}{\rm{.}}\left( {a,b} \right) \times {\rm{L}}{\rm{.C}}{\rm{.M}}{\rm{.}}\left( {a,b} \right) = a \times b\]
Substituting \[{\rm{H}}{\rm{.C}}{\rm{.F}}{\rm{.}}\left( {a,b} \right) = 5\] and \[a \times b = 750\] in the equation, we get
\[ \Rightarrow 5 \times {\rm{L}}{\rm{.C}}{\rm{.M}}{\rm{.}}\left( {a,b} \right) = 750\]
This is a linear equation. We will simplify this linear equation to find the value of the L.C.M.
Dividing both sides of the equation by 5, we get
\[ \Rightarrow \dfrac{{5 \times {\rm{L}}{\rm{.C}}{\rm{.M}}{\rm{.}}\left( {a,b} \right)}}{5} = \dfrac{{750}}{5}\]
Simplifying the expression, we get
\[\therefore \text{L}\text{.C}\text{.M}\text{.}\left( a,b \right)=150\]
Therefore, we get the L.C.M. of the two numbers as 150.

Note: We used the terms H.C.F. and L.C.M. in the solution. The H.C.F., or highest common factor of the numbers \[a\] and \[b\] is the product of the common prime factors of \[a\] and \[b\] with the lowest powers.
The L.C.M., or lowest common multiple of the numbers \[a\] and \[b\] is the product of the prime factors of \[a\] and \[b\] with the greatest powers.