Question

Find the H.C.F and L.C.M of the pairs of integers and verify that $L.C.M\left( a,b \right)\times H.C.F\left( a,b \right)=a\times b$ for 125 and 55

Answer 1

Hint: First of all find the prime factorization of 125 and 55 and then find the H.C.F and L.C.M of 125 and 55. H.C.F of the two numbers is the multiplication of the factors which are common in both 125 and 55 and L.C.M is found by multiplying the H.C.F along with the uncommon factors of both the numbers. Then substitute the value of H.C.F and L.C.M in the given equation $L.C.M\left( a,b \right)\times H.C.F\left( a,b \right)=a\times b$ to verify the equation.

Complete step-by-step answer:
We have given a pair of integers 125 and 55 and we have to find the H.C.F and L.C.M of them.
First of all find the prime factorization of 125 and 55.
Prime factorization of:
$\begin{align}
  & 125=5\times 5\times 5 \\
 & 55=5\times 11 \\
\end{align}$
As you can see from the above prime factorization of 125 and 55, 5 is a factor which is common in both 125 and 55 and we know that H.C.F is the highest common factor or the factors which are common in both the numbers as only 5 is common in both the numbers so H.C.F is equal to 5.
We know that L.C.M is the multiplication of H.C.F with uncommon factors of the two numbers. H.C.F is 5 and uncommon factors of both the numbers are:
$\begin{align}
  & 5\times 5\times 11 \\
 & =275 \\
\end{align}$
Now, multiplying the above result by 5 we get the L.C.M of the two numbers.
$275\times 5=1375$
Hence, the L.C.M of 125 and 55 is 1375 and H.C.F is 5.
Now, we have to verify the following equation,
$L.C.M\left( a,b \right)\times H.C.F\left( a,b \right)=a\times b$
In the above equation, the values of a and b are 125 and 55 respectively and L.C.M and H.C.F us equal to 275 and 5 respectively so substituting these values in the above equation we get,
$\begin{align}
  & L.C.M\left( 125,55 \right)\times H.C.F\left( 125,55 \right)=125\times 55 \\
 & \Rightarrow 1375\times 5=125\times 55 \\
\end{align}$
Multiplying the left hand and right hand side of the above equation we get,
$6875=6875$
In the above equation, L.H.S is equal to R.H.S which means that L.C.M and H.C.F are satisfying the given equation.

Note:The take home message that you can get from this problem is that if we have two numbers and if we know the H.C.F of these two numbers then L.C.M can be calculated by using the equation given in the question i.e. $L.C.M\left( a,b \right)\times H.C.F\left( a,b \right)=a\times b$. You might argue that we can easily find the L.C.M of 125 and 55 but suppose if you have given numbers which are 4 digit or 5 digit long then this equation will minimize your calculations.