Question

For the following pairs of numbers, verify the property:
Product of the number \[ = \] Product of their H.C.F. and L.C.M.
\[25,65\]

Answer 1

Hint: In this question, we have to find the H.C.F. and the L.C.M. of the given two numbers and produce them. Then multiply the given two numbers and compare the answers to get the desired proof.

Complete step-by-step solution:
To find the H.C.F., follow the steps as given below:
\[\begin{array}{*{20}{c}}
  {{\text{ }}5\left| \!{\underline {\,
  {25} \,}} \right. } \\
  {5\left| \!{\underline {\,
  5 \,}} \right. } \\
  {{\text{ }}\left| \!{\underline {\,
  1 \,}} \right. }
\end{array}\]
First, we take the factors of \[25\]. \[25\] is divisible by \[5\] then \[5\] is yet divisible by \[5\] again.
To find the H.C.F. of \[65\], follow the steps given below:
\[\begin{array}{*{20}{c}}
  {{\text{ }}5\left| \!{\underline {\,
  {65} \,}} \right. } \\
  {13\left| \!{\underline {\,
  {13} \,}} \right. } \\
  {{\text{ }}\left| \!{\underline {\,
  1 \,}} \right. }
\end{array}\]
\[65\] is divisible by \[5\], \[13\] times. \[13\] is divisible by \[13\] itself because it is a prime number.
Now,
The common factor in both the numbers is \[5\].
Therefore, H.C.F. \[ = 5\]
To find L.C.M., we take all the factors of the numbers, taking the common factor only once.
Factors of \[25 = 5 \times 5\]
Factors of \[65 = 13 \times 5\]
Now, the common factor in both the numbers is \[5\], so we take \[5\] only once combining both the numbers.
L.C.M. \[ = 5 \times 5 \times 13\]
Multiplying the terms, we get;
\[ \Rightarrow \]L.C.M. \[ = 325\]
Now,
The product of H.C.F. and L.C.M.:
H.C.F. \[ \times \] L.C.M. \[ = \]\[325 \times 5\]
\[ \Rightarrow \]The product of H.C.F and L.C.M. \[ = 1625\]
Now, we find the product of the two numbers:
The product of the two numbers\[ = 25 \times 65\]
\[ \Rightarrow \] The product of the two numbers \[ = 1625\]
Since the product of the L.C.M. and H.C.F. is equal to the product of the given numbers, it is proved that;
Product of the number \[ = \] Product of their H.C.F. and L.C.M.
Hence verified.

Note: We have to mind that, H.C.F. Also known as the highest common factor is found by multiplying all the factors of the given numbers which appear in both the numbers. While, L.C.M. also known as the lowest common multiple is found by multiplying the factors which appear in any one of the either list.